Final answer:
Talia's game involves calculating the expected value using probabilities of possible outcomes. With a 40% chance to make each shot, the expected value of playing the game once is $4.00. Since the game costs $5 to play, Talia would, on average, lose $1 per game.
Step-by-step explanation:
The student's question relates to the concepts of probability and expected value in mathematics. Talia wants to play a basketball game where the cost and rewards vary depending on the outcome of her shots. She has a 40% chance of making any given shot. This scenario is a classic problem in probability theory where the student needs to calculate the expected value of playing the game to determine if it is favorable to play in the long term. Expected value is calculated by multiplying the probabilities of the outcomes by their respective rewards or costs and summing these products.
For calculating the expected value for Talia's game, the outcomes are simply:
- Missing both shots, which has a probability of 0.6 × 0.6 = 0.36 and a return of $0.
- Making one shot, which has a probability of 2 × 0.4 × 0.6 = 0.48 and a return of $5.
- Making both shots, which has a probability of 0.4 × 0.4 = 0.16 and a return of $10.
The expected value would then be calculated as follows: (0.36 × $0) + (0.48 × $5) + (0.16 × $10) = $0 + $2.40 + $1.60 = $4.00.
Since Talia pays $5 to play and the expected value of playing is $4.00, she would, on average, lose $1 each time she plays the game.