Two-Box Lottery
At a school festival, a booth runs a lottery with two boxes. Box A contains 10 tickets, of which 3 are winning tickets. Box B contains 8 tickets, of which 1 is winning. A player first rolls a fair six-sided die. If the die shows 1 or 2, the player draws one ticket at random from Box A; otherwise the player draws one ticket at random from Box B. A ticket costs 200 yen, and a winning ticket pays 1,000 yen. The booth organizer wants to know whether the game is profitable on average, and also wants to understand the information revealed when a player wins. Tickets are drawn without replacement within a box, but each player uses a freshly restocked box, so draws by different players are independent.
1. Find the probability that a player wins a prize. Show the tree calculation.
Read the worked answer
Let A be the event of drawing from Box A and W be winning. P(A)=2/6=1/3 and P(B)=4/6=2/3. Also P(W|A)=3/10 and P(W|B)=1/8. By the law of total probability, P(W)=P(A)P(W|A)+P(B)P(W|B)=(1/3)(3/10)+(2/3)(1/8)=1/10+1/12=6/60+5/60=11/60≈0.1833.
2. Compute the expected net gain per play for the player and decide whether the booth makes a profit in the long run.
Read the worked answer
The expected prize paid to a player is 1000·P(W)=1000·11/60=11000/60=550/3≈183.33 yen. The net gain per play is expected prize minus cost: 550/3-200=(550-600)/3=-50/3≈-16.67 yen. So the player loses about 16.67 yen per play on average, and the booth gains the same amount per play in the long run. Using only the probability of winning, or only the 1,000-yen prize, would miss the 200-yen cost.
3. Given that a player won, what is the probability that the player drew from Box A? Interpret the result.
Read the worked answer
Given a win, P(A|W)=P(A∩W)/P(W)=(1/3·3/10)/(11/60)=(1/10)/(11/60)=6/11≈0.545. Before the draw, P(A)=1/3≈0.333. A winner is therefore more likely to have come from Box A, because Box A has the higher winning proportion (3/10 versus 1/8). This is a conditional-probability statement, not a claim that most players draw from Box A; in fact only one-third do.