Question

A box has 10 tickets in it, two of which are winning tickets. You draw a ticket at random. If it's a winning ticket, you win. If not, you get another chance, as follows: your losing ticket is replaced in the box by a winning ticket (so now there are 10 tickets, as before, but 3 of them are winning tickets). You get to draw again, at random. Which of the following are legitimate methods for using simulation to estimate the probability of winning? I. Choose, at random, a two-digit number. If the first digit is 0 or 1, you win on the first draw; If the first digit is 2 through 9, but the second digit is 0, 1, or 2, you win on the second draw. Any other two-digit number means you lose. Il. Choose, at random, a one-digit number. If it is 0 or 1, you win. If it is 2 through 9, pick a second number. If the second number is 8, 9, or 0, you win. Otherwise, you lose. Ill. Choose, at random, a one-digit number. If it is 0 or 1, you win on the first draw. If it is 2, 3, or 4, you win on the second draw; If it is 5 through 9, you lose.

Answer 1

Only Method I is a legitimate simulation to estimate the winning probability correctly, as it accurately represents the chances of winning both in the initial draw and after a non-winning ticket is replaced with a winning one.

Step-by-step explanation:

The student is asking about methods to estimate the probability of winning a game by drawing tickets from a box using simulation. The first step in the process is to understand how many winning tickets there are and the process of the draw. There are two winning tickets out of ten initially. If the first draw is not a winning ticket, another winning ticket is added, and thus the second draw has 3 winning tickets out of ten. The simulation methods provided are trying to replicate these conditions in a simpler form using random number generation.

Out of the three proposed simulation methods:

Method I is a legitimate simulation as it correctly represents the chances of winning on the first draw (20% or 2 out of 10) and then, if necessary, the second draw (30% or 3 out of 10).

Method III is incorrect as it shifts the probability of winning in the second draw to winning numbers that were not winners in the first draw.

Therefore, only Method I properly simulates the described scenario of drawing winning tickets from a box with the given replacement rule.