In Example 16 and in the previous exercise, we used random numbers to estimate the probability of winning at least $20 in the game. Recall that to do so, for each repetition we simply recorded a Yes or No, pending whether or not we won at least $20 on the given repetition. At the end of the series of repetitions, we essentially determined the proportion of Yes answers in our sequence of repetitions. Simulation can actually be used to answer many more questions than just those about probability. For example, suppose that rather than estimating the probability of winning at least $20, we wanted to estimate the expected winnings from playing the game. We need to make two minor adjustments to the approach used in the example. Instead of recording Yes or No for each repetition, record the actual dollar amount won. At the end of the series of repetitions, simply calculate the average winnings across all repetitions. This provides an estimate for the amount we can expect to win by playing the game.
a. Beginning at line 5 of the table of random digits, carry out 20 repetitions and report the average winnings.
b. How does the estimated average winnings compare with the price to play the game?