Name: Renee

Who is asking: Student

Level: Secondary

Question:

Can you help me figure out how to solve this problem? Mistie, Tammy, and Jennifer audition for parts in a big-budget remake of Bedtime for Bonzo. On the basis of their past experience and the caliber of competition they face, Mistie has a 40% chance of being hired, Tammy has a 50% chance, and Jennifer has a 30% chance. If exactly two of the three are cast, what is the probability that Mistie was rejected? Thank you!

Hi Renee,
I think this is a question on CONDITIONAL probability.
Consider two events: A=" Exactly two of the three are cast" and
B=" Mistie was rejected"
Then we are interested in the event P(B|A) (conditional probability of event B
given that event A holds). The formula for calculating a conditional probability
is as follows:

P(B|A)=P(A and B)/P(A).
Now we can calculate all probabilities.
Event {A and B} - Exactly two are cast AND Mistie was rejected, that is, -
Mistie rejected AND Tammy hired AND Jennifer hired. The probability of Mistie being
rejected is P(B) = 1- P( Mistie was hired) = 1- 0.4 = 0.6, all other probabilities
are given. We can assume that everything independent, so
P(A and B) = (0.6)*(0.5)*(0.3) = 0.09.
Now event A - Exactly two are cast. There are 3 outcomes for this event:

Mistie rejected AND Tammy hired AND Jennifer hired,

Mistie hired AND Tammy rejected AND Jennifer hired,

Mistie hired AND Tammy hired AND Jennifer rejected,
By the same idea we used before,

P( Mistie rejected AND Tammy hired AND Jennifer hired )=(0.6)*(0.5)*(0.3)=0.09

P( Mistie hired AND Tammy rejected AND Jennifer hired )=(0.4)*(0.5)*(0.3)=0.06

P( Mistie hired AND Tammy hired AND Jennifer rejected
)=(0.4)*(0.5)*(0.7)=0.14
Then P(A)= sum of these probabilities = 0.09+0.06+0.14 = 0.29.

Now, by the formula of conditional probability P(B|A)= 0.09/0.29=9/29=0.3103.

With kind regards, Andrei