Answer: Therefore, the expected number of times the player will get audited in 5 turns is 1.25.
Step-by-step explanation:
To find the expected number of times the player will get audited in 5 turns of rolling the dice, we need to calculate the probability of getting audited on each turn and then sum up those probabilities.
Let's consider each possible sum and its corresponding probability:
Sum of 7: There are six combinations that result in a sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). Each combination has a probability of 1/36 since there are 36 possible outcomes when rolling two dice. Therefore, the probability of getting audited with a sum of 7 is (6/36) = 1/6.
Sum of 11: There are two combinations that result in a sum of 11: (5, 6) and (6, 5). Each combination has a probability of 1/36. So, the probability of getting audited with a sum of 11 is (2/36) = 1/18.
Sum of 12: There is only one combination that results in a sum of 12: (6, 6). It has a probability of 1/36. Hence, the probability of getting audited with a sum of 12 is (1/36).
For any other sum, the player avoids taxes.
Now, let's calculate the expected number of times the player will get audited in 5 turns. Since each turn is independent, we can multiply the probability of getting audited in a single turn by the number of turns (5).
Expected number of audits = (Probability of getting audited on each turn) * (Number of turns)
= [(1/6) + (1/18) + (1/36)] * 5
= (6/36 + 2/36 + 1/36) * 5
= (9/36) * 5
= (1/4) * 5
= 5/4
= 1.25