a If the expected payoff value is greater than 0, then we should play.
B
b Check if the expected payoff values are greater than 0 after changing the rules of the lottery.
A
a Yes, see solution.
B
b No, because the expected payoff is still greater than 0.
Practice makes perfect
a We are told that there is a lottery in which we should match 5 numbers out of 31 possible outcomes in order to win. To decide whether we should play or not we use the expected payoff value, which is defined as the expected value minus the ticket cost.
Expected Payoff = Expected Value - Ticket Cost
We are given that a ticket for this lottery costs $1, so let's focus on calculating the expected value. The expected value EV will be the probability of winning the lottery P_w multiplied by the value of the prize, $1 000 000, plus the probability of losing P_l multiplied by $0.
EV=P_w* 1 000 000 + P_l* 0 ⇕
EV=P_w* 1 000 000
Next, let's rewrite the probability of winning the lottery. Since we have only 1 set of numbers that makes us win, the number of favorable outcomes is 1. The number of all possible outcomes will be a number of all possible combinations of 5 elements from the set of 31 numbers.
EV=1/_(31)C_5* 1 000 000
Notice that we need to consider combinations because we are told that any sequence of the winning numbers will win the prize. Now, let's simplify the expression using the Combination Formula to evaluate the expected value.
The expected value of this lottery is approximately $5.89. Finally, let's subtract the ticket cost, $1 from the expected value.
Expected Payoff= $5.89- $1=$4.89
Since the expected payoff for the lottery is greater than 0, we should play.
b Now, we are asked to decide whether we want to play the lottery if there were changes to the prizes and rules. Let's consider the given scenarios one at a time.
If the Prize Money Increased
First, we want to see what the expected payoff would be if the winnings increased to $5 million. Since the probability of winning is the same as in Part A, we can directly calculate the expected value of the lottery after this change.
The expected value of this lottery is approximately $29.43. Let's subtract the ticket cost, $1 from the expected value to find the expected payoff.
Expected Payoff= $29.43- $1 =$28.43
Since the expected payoff for this lottery is still greater than 0, our decision should also be to play.
If the Prize Money Decreased and the Chosen Numbers Increased
Next, let's assume that the prize is decreased to $0.5 million but we are choosing from 21 numbers instead of 31. Let's evaluate the expected value of the lottery with these changes values.
The expected value is approximately $24.57. Finally, let's subtract the ticket cost, $1 from the expected value.
Expected Payoff= $24.57- $1 =$23.57
The expected payoff for this lottery is still greater than 0, so the decision to play stays the same.
