Question

A carnival game allows a group of players to each draw and keep a marble from a bag. The bag contains 5 gold marbles, 25 silver marbles, and 70 red marbles.
A player wins a large prize for drawing a gold marble and a small prize for drawing a silver marble. There is no prize for drawing a red marble.

At the start of the game, the probability of winning a large prize is 0.05 and the probability of winning a small prize is 0.25.

1. Suppose that the first player draws a silver marble and wins a small prize. What is the probability that the second player will also win a small prize?

2. If a group of four plays the game one at a time and everyone wins a small prize, which player had the greatest probability of winning a large prize?

3. How could the game be made fair for each player? That is, how could you change the game so that each player has an equal chance of winning a prize?

Answer 1

5 gold marbles,
25 silver marbles, and
70 red marbles.
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100 total marbles

large prize: drawing a gold marble
small prize: drawing a silver marble.

At the start of the game,

probability of winning a large prize = positive outcoumes / total possible outcomes = 5 gold marbles / 100 total marbles =  0.05

probability of winning a small prize = positive outcomes / total possible outcomes = 25 silver marbles / 100 total marbles = 0.25.

1. Suppose that the first player draws a silver marble and wins a small prize. What is the probability that the second player will also win a small prize?

Answer:

numer of silver marbles / number of total marbles = (25 -1 ) / (100 - 1) = 24 / 99 ≈ 0.24

2. If a group of four plays the game one at a time and everyone wins a small prize, which player had the greatest probability of winning a large prize?

Answer:

First player: 0.25

Second player: 5 gold marbles / ( 100 - 1) total marbles = 5 /99 ≈ 0.0505

Third player: 5 gold marbles / (99 - 1) total marbles = 5 / 98 ≈ 0.051

Fourth player: 5 gold marbles / ( 98 - 1) total marbles = 5 / 97 ≈ 0.0515

So, the probability of winning a big prize increases as more balls different of gold marbles are extracted from the bag, and so, in this case, the fourth player has a greater chance to win a large prize.

3. How could the game be made fair for each player? That is, how could you change the game so that each player has an equal chance of winning a prize?

Answer: All the players would have equal chance of winning a prize if the balls were replaced in the bag after each play.