SOLUTION: Suppose a math class contains 31 students, 10 females (six of whom speak French) and 21 males (three of whom speak French). Compute the probability that a randomly selected student

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Question 1048712: Suppose a math class contains 31 students, 10 females (six of whom speak French) and 21 males (three of whom speak French). Compute the probability that a randomly selected student is female, given that the student speaks French.
Answer by Theo(13342) About Me  (Show Source):
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the probability that a randomly selected person from the class is a woman would be 10 / 31 because 10 out of the 31 members of the class are women.

the probability that a randomly selected person from the class is a woman and speaks french is 6 / 31 because 6 out of the 31 members of the class are women that speak french.

given that the person speaks french, what is the possibility that the randomly selected person speaks french.

there are 9 members of the class that speak french and 6 of them are women, so the probability that the randomly selected person is a woman given that the person speaks french is 6/9 = 2/3.

you can draw a table that depicts this situation.

the table would look like this: 



                         men             women           total

speaks french             3                6               9
doesn't speak french      18               4               22
total                     21               10              31


you can see from the table that there are a total of 9 people who speak french and 6 of them are women.

you pick a random person from the group and that person speaks french.
that person has to be 1 of the 9 who speak french.
6 women speak french, so the probably that the person is a woman who speaks french is 6/9 = 2/3.

the formula for p(a given b) = p(a and b) / p(b).

if you let a be the event that the person is a woman and you let b be the event that the person speaks french, then you get the following:

p(a and b) is the probability that the person speaks french and is a woman.
since 6 of the people in the group are women that speak french, then the probability of (a and b) becomes 6/31.

the probability that a person speaks french is 9/31 because 9 people in the group out of 31 speak french.

p(a given b) = p(a and b) / p(b) becomes p(a given b) = (6/31) / (9/31) which is the same as p(a given b) = (6/31) * (31/9) which becomes p(a given b) = 6/9 which can be simplified to p(a given b) = 2/3.

since a is the event that the person is a woman and b is the events that the person speaks french, then p(a given b) = 2/3 means the probability that the person is a woman given that the person speaks french is equal to 2/3.