Question

1. you bet! you roll a die. if it comes up a 6, you win $100. if not, you get to roll again. if you get a 6 the second time, you win $50. if not, you lose. 17. get it right? students called

Explanation:

Step1: Define Probabilities

A die has 6 faces. Probability of rolling a 6: \( P(6_1) = \frac{1}{6} \). Probability of not rolling a 6: \( P(\text{not }6_1) = \frac{5}{6} \). For the second roll, probability of 6: \( P(6_2) = \frac{1}{6} \), not 6: \( P(\text{not }6_2) = \frac{5}{6} \).

Step2: Calculate Expected Value (EV)

EV = (Win $100$ × \( P(6_1) \)) + (Win $50$ × \( P(\text{not }6_1) \times P(6_2) \)) + (Win $0$ × \( P(\text{not }6_1) \times P(\text{not }6_2) \))
EV = \( 100 \times \frac{1}{6} + 50 \times \frac{5}{6} \times \frac{1}{6} + 0 \times \frac{5}{6} \times \frac{5}{6} \)
Calculate each term:
\( 100 \times \frac{1}{6} = \frac{100}{6} \approx 16.67 \)
\( 50 \times \frac{5}{36} = \frac{250}{36} \approx 6.94 \)
Sum: \( \frac{100}{6} + \frac{250}{36} = \frac{600 + 250}{36} = \frac{850}{36} \approx 23.61 \)

Answer:

The expected value of the bet is approximately \(\$23.61\).