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# S2 clarification on how to work out probability question - pints of milk combinations watch

1. You are given that the probability of a pint of milk containing less than 1 pint = 0.005. A crate contains 24 bottles.

First part of question I can do ok.

Find the probability that no bottles contain less than 1 pint of milk.

Probability that a bottle does NOT contain less than 1 pint = 1 - 0.005 = 0.995.

prob no bottles in crate contain less than 1 pint = 0.995^24 = 0.8865. This answer agrees with my answer sheet.

But I want to check I am doing the second part of the question correctly as it is trickier.

Find the probability that at most 1 bottle contains less than 1 pint of milk.

Well this probability must be higher than ALL 24 NOT less than 1 pint because one of them can or can not.

So is it worked out like this:

0.995^23 * 1

1 - meaning includes both options - ie could be less than 1 pint but doesn't have to be?

0.995^23 = 0.8911

but book answer says 0.9936 - so mine must be wrong?

How do I do this?

[Just in case it is important this is part of a Normal distribution problem with mean = 1.036 and sd=0.014. Question says you can use 0.005 as exact number for prob for less than 1 pint].
2. The probability of a bottle containing less than 1 pint of milk is distributed binomially.

(Original post by acomber)
Find the probability that at most 1 bottle contains less than 1 pint of milk.
This is the equal to the probability no bottles contain less than 1 pint of milk + the probability of 1 bottle containing less than 1 pint of milk.

Remember that there are 24C1 ways for 1 of the 24 bottles to contain less than 1 pint of milk.
3. (Original post by SherlockHolmes)
The probability of a bottle containing less than 1 pint of milk is distributed binomially.

This is the equal to the probability no bottles contain less than 1 pint of milk + the probability of 1 bottle containing less than 1 pint of milk.

Remember that there are 24C1 ways for 1 of the 24 bottles to contain less than 1 pint of milk.
so its:

0.995^24 + 24 C 1 x 0..5 x 0.995^23

of course.

which is really

24 C 0 x 0.995^24

+

24 C 1 x 0..005^1 x 0.995^23

Thanks.

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