Answer:
To determine how much tickets should cost for Jennifer's game to make a profit, we need to consider the probabilities of each outcome and the associated payouts.
Explanation:
1. Probability of drawing a red card: There are 26 red cards in a standard 52-card deck, so the probability of drawing a red card is 26/52 or 1/2.
2. Probability of spinning a 1 on the spinner: There are 5 equal slices on the spinner, so the probability of spinning a 1 is 1/5.
3. Probability of drawing a black card: Since there are 26 black cards in a standard deck, the probability of drawing a black card is also 1/2.
4. Probability of spinning a 2 or 3 on the spinner: There are 5 equal slices on the spinner, and 2 of them correspond to spinning a 2 or 3. Therefore, the probability of spinning a 2 or 3 is 2/5.
Now let's calculate the expected payouts for each outcome:
- Winning $30 (drawing a red card and spinning a 1): Probability = (1/2) * (1/5) = 1/10. Expected payout = (1/10) * $30 = $3.
- Winning $20 (drawing a black card and spinning a 2 or 3): Probability = (1/2) * (2/5) = 1/5. Expected payout = (1/5) * $20 = $4.
- Losing: The probability of losing is 1 - (1/10 + 1/5) = 1 - 3/10 = 7/10.
To make a profit, the expected payouts should be less than the cost of the tickets. Since the probabilities of winning are relatively low, we can assume that most players will lose. Therefore, the expected payout from losses should be significantly lower than the cost of the tickets.
Let's assume the cost of the tickets is $x. To ensure a profit, the expected payout from losses should be less than $x.
Expected payout from losses = (7/10) * $0 = $0.
To guarantee a profit, the cost of the tickets should be greater than $0. Therefore, tickets should cost more than $0 for the amusement park to make a profit. However, it is important to consider additional factors such as operational costs, desired profit margin, and competitiveness in the market when setting the ticket price.