Could Anyone Help With This Math Question on Probability?

Question

Four friends want to play a game. In order to decide who starts first, two proposals are made.

Kristal proposes that they assign everyone a number and take turns rolling a six-sided die. The player whose number comes up would start the game, and if a 5 or 6 comes up, they roll again.

Clay proposes that they flip 3 coins. If 3 heads come up, Clark would start. If 2 heads come up, Clay would start. If 2 tails come up, Kortney would start. And if 3 tails come up, Kristal would start.

Which of the proposals gives everyone a fair chance of starting the game? Explain your reasoning.

Answer 1

The first option is definitely the best option. Each has a 1/4 probability of starting the game. It says that if a 5 or 6 is rolled, they will just keep trying until they get a number between 1 and 4. Since each roll is independent and they're disregarding the 5 and 6 rolls, it's effectively as if they never happen and don't exist.

The second plan is very unfair to Clark and Kristal. The only way they could start the game is by getting 3 heads or 3 tails. Here are all the possibilities:

TTT

TTH

THT

HTT

THH

HTH

HHT

HHH

That is a 1/8 (or (1/2)³) probability for each of them. However, Clay and Kortney each get a 3/8 probability of starting the game. This makes 2 players 3 times as likely to start as the other 2. This is far from equal and fair.

Answer 2

Proposal #1: Kristal proposes that they assign everyone a number ( 1 to 4) and take turns rolling a six-sided die. The player whose number comes up would start the game, and if a 5 or 6 comes up, they roll again.

The probability of a getting a number between 1 to 4 is 1/6. Thus everyone has an equal probability of 1/6 of starting the game.

Proposal #2: Clay proposes that they flip 3 coins. If 3 heads come up, Clark would start. If 2 heads come up, Clay would start. If 2 tails come up, Kortney would start. And if 3 tails come up, Kristal would start.

The probability of getting 3 heads = The probability of getting 3 tails = 1/8

The probability of getting 2 heads = The probability of getting 2 tails = 3/8

The 2nd proposal is more biased towards Clay and Kortney.