Probability?

According to genetic theory, the blossom color in the second generationof a certain cross of sweet peas should be red or white in 3:1 ratio. That is, each plant has probability 3/4 of having redblossoms, and the blossomcolor of separate plants are independent.



(a) What is the probability that exactly 6 out of 8 of these plants have red blossoms?



(b) What is the mean number of red-blossomed plants when 80 plants of this type are grown from seeds?



(c) What is the probability of obtaining at least 50 red-blossomed plants whe 80 plants are grown frm seeds?
Probability?
Let X be the number of plants with red blossoms. X has the binomial distribution with n = 8 trials and success probability p = 0.75



In general, if X has the binomial distribution with n trials and a success probability of p then

P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)

for values of x = 0, 1, 2, ..., n

P[X = x] = 0 for any other value of x.



The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.

Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.



X ~ Binomial( n , p )



the mean of the binomial distribution is n * p = 6

the variance of the binomial distribution is n * p * (1 - p) = 1.5

the standard deviation is the square root of the variance = √ ( n * p * (1 - p)) = 1.224745



The Probability Mass Function, PMF,

f(X) = P(X = x) is:



P( X = 0 ) = 1.525879e-05

P( X = 1 ) = 0.0003662109

P( X = 2 ) = 0.003845215

P( X = 3 ) = 0.02307129

P( X = 4 ) = 0.08651733

P( X = 5 ) = 0.2076416

P( X = 6 ) = 0.3114624 ← answer

P( X = 7 ) = 0.2669678

P( X = 8 ) = 0.1001129



== -- == -- == -- == -- ==



b)



80 * 0.75 = 60



== -- == -- == -- == -- ==



c)



To use the normal approximation to the binomial you must first validate that you have more than 10 expected successes and 10 expected failures. In other words, you need to have n * p %26gt; 10 and n * (1-p) %26gt; 10.



Some authors will say you only need 5 expected successes and 5 expected failures to use this approximation. If you are working towards the center of the distribution then this condition should be sufficient. However, the approximations in the tails of the distribution will be weaker especially if the success probability is low or high. Using 10 expected successes and 10 expected failures is a more conservative approach but will allow for better approximations especially when p is small or p is large.



In this case you have:

n * p = 80 * 0.75 = 60 expected success

n * (1 - p) = 80 * 0.25 = 20 expected failures



We have checked and confirmed that there are enough expected successes and expected failures. Now we can move on to the rest of the work.



If Xb ~ Binomial(n, p) then we can approximate probabilities using the normal distribution where Xn is normal with mean μ = n * p, variance σ2 = n * p * (1-p), and standard deviation σ



Xb ~ Binomial(n = 80 , p = 0.75 )

Xn ~ Normal( μ = 60 , σ2 = 15 )

Xn ~ Normal( μ = 60 , σ = 3.872983 )



I have noted two different notations for the Normal distribution, one using the variance and one using the standard deviation. In most textbooks and in most of the literature, the parameters used to denote the Normal distribution are the mean and the variance. In most software programs, the standard notation is to use the mean and the standard deviation.



The probabilities are approximated using a continuity correction. We need to use a continuity correction because we are estimating discrete probabilities with a continuous distribution. The best way to make sure you use the correct continuity correction is to draw out a small histogram of the binomial distribution and shade in the values you need. The continuity correction accounts for the area of the boxes that would be missing or would be extra under the normal curve.



P( Xb %26lt; x) ≈ P( Xn %26lt; (x - 0.5) )

P( Xb %26gt; x) ≈ P( Xn %26gt; (x + 0.5) )

P( Xb ≤ x) ≈ P( Xn ≤ (x + 0.5) )

P( Xb ≥ x) ≈ P( Xn ≥ (x - 0.5) )

P( Xb = x) ≈ P( (x - 0.5) %26lt; Xn %26lt; (x + 0.5) )

P( a ≤ Xb ≤ b ) ≈ P( (a - 0.5) %26lt; Xn %26lt; (b + 0.5) )

P( a ≤ Xb %26lt; b ) ≈ P( (a - 0.5) %26lt; Xn %26lt; (b - 0.5) )

P( a %26lt; Xb ≤ b ) ≈ P( (a + 0.5) %26lt; Xn %26lt; (b + 0.5) )

P( a %26lt; Xb %26lt; b ) ≈ P( (a + 0.5) %26lt; Xn %26lt; (b - 0.5) )



In the work that follows Xb has the binomial distribution, Xn has the normal distribution and Z has the standard normal distribution.



Remember that for any normal random variable Xn, you can transform it into standard units via: Z = (Xn - μ ) / σ



P( Xb ≥ 50 ) =



80

∑ P(Xb = x) = 0.995418

x = 50



≈ P( Xn ≥ 49.5 )

= P( Z ≥ ( 49.5 - 60 ) / 3.872983 )

= P( Z ≥ -2.711088 )

= 0.9966469
Reply:The Binomial distribution has the probability

function P(x=k)= nCr p^r (1-p)^(n-r)

k=0,1,2,.....,n

where nCr = n! / r! (n-r)!

a) n=8, x=6, p=3/4 or 0.75

0.3115



b)mean of the Binomial distribution is np = 80(3/4) =60



c)n=80, x=50 through 80, p=3/4 or 0.75

You may use Normal approximation, but let us directly evaluate it.

P(x %26gt;=50) =0.9954

I have included a source that would help you compute these probabilities. If you have access to a Binomial table, you may use it.

Which Bath & Body Works Signature Collection fragrance do you like the best? need male and female opinions?

here are the collection



Bath %26amp; Body Works Signature Collection



* Black Raspberry Vanilla

* Blackberry Amber

* Blushing Cherry Blossom

* Brown Sugar %26amp; Fig

* Cherry Blossom

* Chocolate Amber

* Coco Cabana

* Coconut Lime Verbena

* Cool Citrus Basil

* Country Apple

* Creamy Coconut

* Cucumber Melon

* Dancing Waters

* Enchanted Orchid

* Exotic Coconut

* Flowering Herbs

* Fresh Pineapple

* Freshwater Cucumber

* Gardenia

* Green Clover %26amp; Aloe

* Green Tea %26amp; Cucumber

* Irresistible Apple

* Japanese Cherry Blossom

* Juniper Breeze

* Lemongrass Sage

* Magnolia Blossom

* Mango Mandarin

* Midnight Pomegranate

* Moonlight Path

* Night Blooming Jasmine

* Pearberry

* Pink Grapefruit

* Plumeria

* Rice Flower %26amp; Shea

* Sea Island Cotton

* Seaspray

* Sensual Amber

* Sheer Freesia

* Sparkling Peach

* Strawberry Lemonade

* Sun-Ripened Raspberry

* Sweet Cinnamon Pumpkin

* Sweet Pea

* Tropical Passionfruit

* Velvet Tuberose

* Warm Vanilla Sugar

* Water Blossom Ivy

* White Cherry Blossom

* White Tea %26amp; Ginger

* Wild Honeysuckle
Which Bath %26amp; Body Works Signature Collection fragrance do you like the best? need male and female opinions?
1.Cucumber Melon



2.Moonlight Path



3.Sweet Pea
Reply:I ADORE almost all Bath %26amp; Bodyworks products. It would probably be Cool Citrus Basil and Warm Vanilla Sugar. When I need a break from the Signature lines, the Aromatherapy lines work for me too, most notably the Eucalyptus Mint or the Lavender Vanilla.
Reply:you can never go wrong with cucumber melon!

cargo

Will you help pick 2 phrases for M&M's?

We want to give M%26amp;M's to everyone to tell them we are pregnant with baby #2. We get to pick two colors and can pick two phrases. Up to two lines with 8 letters on each line..spaces count. I came up with all of these. Pick 2 favorites or come up with your own!

Bun In The Oven, New Baby Coming, Baby On The Way, New Baby (last name), Baby #2 Cooking, New Big Brother, New Baby Sweet Pea, 6 Months To Go, Due In (Month Due)
Will you help pick 2 phrases for M%26amp;M's?
Baby on the way



Baby #2 cooking



What colors are you doing??

Lavender and pale green

Peach and Yellow



Blue and pink are too overdone!
Reply:Thanks. Best of luck to you and your family!!! Congratulations!



:) :) :) Report It
Reply:I like the baby #2 cooking. Congratulations by the way
Reply:Bun in the Oven and new baby(last name)-colors-yellow and orange...
Reply:I like baby on the way or new baby (last name). Congrats!
Reply:AI like New Baby (last name) and New Baby Coming.



Congratulations!!!
Reply:i agree w/ #25
Reply:I like New Baby (last name) and Due in (Month due) with pink and blue since you're not sure about the sex yet.



Congratulations on Baby #2!
Reply:I like baby on the way, and Due in (month due)....

cute idea, wished I could of thought of that.....=]

What do we want to eat tonight?

Smothered Pork Steak, Baked Potato Loaded, Sweet Peas, Corn bread, and Chocolate Pudding

or (tea, soda or water to drink)...with either

Chicken Stirfry, egg drop soup, spring roll, and flan
What do we want to eat tonight?
I would gladly consume the smotheredf pork steak, baked tater but only a little butter and chives, please, peas, cornbread, and pudding, too. I'll take iced tea, unsweetened with a lemon. Thanks for inviting me to dinner...
Reply:My Daughter and I just Had Mac and Cheese...she drank milk and I had a Sam Adams....... and for desert we are having J E L L O.......
Reply:Chicken stirfry, egg drop soup, spring roll and flan with water!
Reply:Chicken Stirfry, egg drop soup, spring roll, and flan
Reply:I'll go chicken stir fry.I hate pork steak anyway it's cooked.Tea as the beverage please.
Reply:Personally, I LOVE stirfry..... :)
Reply:I have everything except the smothered pork steak and the egg drop soup, and tea to drink please.
Reply:smothered pork steak....yum



finally got that homemade meal tonight and I had a choice of two favorites: spaghetti or pot roast. either feast or famine here... but it is all good.
Reply:Chicken stirfry, egg drop soup, spring roll, and flan
Reply:Ha ha, I just had the chicken stirfry, but just with an eggroll. The egg drop soup would be nice...could you bring me some?
Reply:My father just cooked some pork ribs and rice
Reply:Popcorn, candy %26amp; soda. Mmmmm......
skates

What's for dinner?

I am making BBQ chicken,mac-n-cheese,sweet peas and baked potaoes.
What's for dinner?
I am actually one of the few Americans who is looking to put on some pounds. So I had what I call "Supper" and, "Second Supper" today lol.



First I had two Egg sandwitches.

4 eggs mixed in a bowl with milk

mexican cheese sprinkled ontop

put inbetween two pieces of toast. YUMMY!



And for second supper I had:

One cheese breadstick from where else? Pizza Hut!

And two Cinamon Waffles.



Maybe I'll come over to your house tomorrow afternoon and have your left-overs. Your meal sounds pretty dang good!
Reply:Home made soft tacos(beef) with a side of home made spanish rice and refried beans. Guacamole and chips and fresh flour tortillas!
Reply:sounds good-I cook romen noodles mixed with broccolli or spinich %26amp; chopped burritos along with additional seasonings. (I'll stir-fry it if I'm not starving %26amp; have patience)
Reply:That sounds good. I had a cheeseburger and fries :)
Reply:I had sausage and eggs, english muffin and coffee. Sometimes I like breakfast for dinner.
Reply:I had macaroni and avocado mixed together and fresh cherries and coca cola. Isn't that weird????
Reply:I had pork shutters and pink tacos.
Reply:We having chicken flavored ramen noddle soup with baked chicken legs and for dessert apple pie and ice-cream.
Reply:Just finished Chinese
Reply:I made baked chicken,pasta salad and Corn on the cob:)



You have been starred!!!
Reply:Italian-breaded oven fried chicken, rice and corn. With chocolate milk, and probably cheerios for the picky baby!
Reply:we had gnocchi for dinner and ice cream for dessert
Reply:Sounds great. What time will you be expecting us?
Reply:mac%26amp; cheese and salad.

Kix for 8m babies?

so my baby refuses baby jar foods, she wants to feed herself. She lives on Cheerios, little pieces of fruits, carrots, sweet peas, corns, etc. Is it safe and good idea to give her Kix? Does that dissolve fast enough? She has 6 teeth.
Kix for 8m babies?
Most small cereals should be fine if she can handle cheerios. I would personally avoid any cereals that have any allergens listed on the ingredient label though - especially peanuts.



Here are some good sites that list foods and ages - it is simply a guideline:

http://www.babysake.com/babyfoodchart.ht...

http://www.babycenter.com/0_baby-food-ba...

http://www.wholesomebabyfood.com/solids....
Reply:If you give her Cheerios, she should be okay w/ Kix. The dissolve much faster.
Reply:They turn to much quicker than cheerios.
Reply:i gave my son little cereals at this time, he ate it good, but there are little puffs they sell for babies that dissolve fast in their mouth.
Reply:Kix does disolve but the shape of them is scary. I would stick with the cheerios. She seems to be fine with what you are giving her, I do not think she is missing anything if you hold off on giving her Kix.
Reply:i never tried KIX, but we did do other cereals. KIX scared me because they were like a ball and i was afraid of choking.

have you tried those gerber puffs, or the freeze dried fruits? MY daughter LOVED those. she still loves the dried fruits. they are in the baby food aisle. also cheeses they are easy to gum and swallow. pastas. bread.
Reply:That is great!!! better off eating real food anyway... I'd just keep experimenting w/ steamed veggies.... I'm babysitting a 9 month old boy who love beans!! his mom brings him tiny ones like navy beans, blackeyes peas, black beans... I'm not sure about the kix - try some for yourself, if they dissolve as fast as cheerios, then they should be ok!



Make sure she's not actually living off these solids though, her breastmilk/formula should be her main diet, and solids should be some snacks for fun and to entertain herself while the rest of the family eats dinner!!!
Reply:There are gerber graduate grain puffs that are like little cereal peices that provide more nutrition than cereals. Kix wouldnt dissolve before a baby could choke on it, they dont know how to properly chew. A babies pimary nutrition before 1 year is formula anyhow, but chocking is a big hazard as is sugars a bad thing for them.
Reply:I think it's fine, but I wouldn't give her too many at once. Even though they turn to mush, if there are too many in her mouth at one time, they could get stuck. My niece used to choke on Goldfish, just because she would take huge handfuls and they would get into this big glob of mush and she would choke...I mean really choke and turn blue. So, just make sure she's not doing the chipmunk thing and storing the kix in her mouth! : )

Moms! baby food question?

is it safe to feed my baby home made baby food? like sweet peas or carrotts, cooked and puread?
Moms! baby food question?
Absolutely, and better for them too. Just make sure you wash the food really well before cooking/puree-ing.



Check out these resources for at-home baby food preparation (there are safe ways and unsafe ways to prepare and store it, so check these sites for tips):

http://www.wholesomebabyfood.com/

http://homecooking.about.com/od/specific...

http://www.homemade-baby-food-recipes.co...



Good luck!
Reply:yes, just make sure you get them fresh or frozen and not in the can. This is really the better way to go the baby will enjoy it more is what some say. My sis in law puriees her babies food. Baby is fine.
Reply:when my kids where baby's i did not buy baby food at the store ever



i bought a food chopper and made my own when i cooked i left alil off to the side and chopped it up for the baby that way i knew what my child was eating
Reply:Yep, it absolutely is. (as long as your baby is 6 months and older) It can actually be better, because you aren't putting any preservatives or additves into your baby's food like baby food companies do.

Here is a great website that can help you with your homemade baby food.

www.wholesomebabyfood.com



I love this website. It is very informative.

Good luck!
Reply:I agree with the Mom's above (and likely below), just one cautionary note: be sure that when your baby gets a little older and moves onto meats and such that you don't put things like pepper/spices on it like you normally would for yourself! The spices will make it too strong for your little one, and may cause some uncomfortable digestion, similar to how breastfeeding mom's typically cut down on spicy foods. Otherwise, go ahead! Mothers for thousands of years had to feed their babies without Gerber to help them along.



It's just my theory, but giving them baby food the way you prepare it versus Gerber may make them more likely to eat the veggies and such that you prepare when they are older kids because it is what they are used to (even if it is no longer smooshed up).
Reply:Of course! Why wouldn't it be?
Reply:yes. Just make sure you feed them the same food for a few days to see if they have a reaction to it. You shouldnt have to add anything to it.



Go to http://www.babycenter.com/baby/babyfeedi... for some tips!

maybelline