Question

The probability that a high school senior wins a prestigious scholarship is 0.13. The probability that a high school senior plays a varsity sport is 0.22 . If the probability that someone plays a varsity sport given that they won the scholarship is 0.55 , then what's the probability that someone wins the scholarship given that they play a varsity sport?

Answer 1

0.325

Explanation:

Let A represent the event: winning a prestigious scholarship
Let B represent the event: plays a varsity sport

Given

• P(A) = 0.13

• P(B) = 0.22

• P(plays a varsity sport given that they won the scholarship)
  = P(B|A) = 0.55

• We are asked to find P(wins the scholarship given play a varsity sport) = P(A|B)

• The formula for conditional probability is
  
  `P(A|B) = (P(A \cap B))/(P(B))\\\\\\P(B|A) = (P(A \cap B))/(P(A))`

• where P(A ∩ B) is the probability of both A and B occurring together

• Rewriting the above two equations we get
  
  `P(A \cap B) = P(A|B)P(B) = P(B|A)P(A)`
  →
  `P(A|B)P(B) = P(B|A)P(A)`
  
• Plugging in known probabilities,
  
  `P(A|B) (0.22) = P(0.55)(0.13)`
  
  `\begin{aligned}P(A|B) & = (0.55 * 0.13)/( 0.22 )\\& = 0.325\end{aligned}`
  

Therefore the probability that someone wins the scholarship given that they play a varsity sport is 0.325