10 of your friends (including you..so 10 altogether) are going to have a draw between all of you. Each person gives $10 (for a total of $100). There will be five, $20 draws. One per week for 5 weeks.
For the first draw, every person gets a ticket. One ticket is picked from 10 tickets.
For the second draw, everyone gets a new ticket. Now, everyone (including the past winner) has two tickets. One ticket is picked from 20 tickets.
For the third draw, everyone gets a new ticket. Now, everyone (including the past winner) has three tickets. One ticket is picked from 30 tickets.
And so on until the draws are finished.
a) One of your friends says that you have a 1 in 10 chance each week to win. Since there are 5 draws, you have a 50% chance of winning at least once. Is your friend correct?
b) What is the expected size of the prize you would win?
c) Is this a fair game? Explain.
in week 1 you have a 1 in 10 chance to win, or 10%. In week 2 you have a 2 in 20 chance to win or 10% .... In week 5 you have a 5 in 50 chance to win, or 10%.
a) The probability that you always lose is .9^5 = .59049 -- so the probability that you win at least once is 1-.59049 = 40.941% -- your friend is incorrect.
b) the expected prize is 5 chances at .1*$20 = 5*.1*$20 = $10.
c) its a fair game. take it from here.
Thank You Very Much!!! Very Much Appreciated!!!
a) To determine if your friend's statement is correct, we need to calculate the probability of winning at least once in the 5 draws.
In each draw, there are a total of 10, 20, 30, 40, and finally 50 tickets respectively. Since each person has one more ticket than the previous draw, it means you have an increasing number of tickets each week.
To calculate the probability of winning at least once in the 5 draws, we can find the probability of not winning in any of the draws and subtract it from 1 (total probability).
In the first draw, you have 1 out of 10 tickets. So the probability of not winning in the first draw is 9/10.
In the second draw, you have 2 out of 20 tickets. So the probability of not winning in the second draw is 18/20.
In the third draw, you have 3 out of 30 tickets. So the probability of not winning in the third draw is 27/30.
And so on, until the fifth draw.
To find the probability of not winning in any of the draws, we multiply all these probabilities:
(9/10) * (18/20) * (27/30) * (36/40) * (45/50) ≈ 0.486
Therefore, the probability of winning at least once in the 5 draws is approximately 1 - 0.486 = 0.514 or 51.4%.
So your friend's statement is incorrect. You actually have a little over 50% chance of winning at least once in the 5 draws, which is slightly higher than 50%.
b) To calculate the expected size of the prize you would win, we need to know the total amount of money you would win and the probability of winning each week.
Since each person contributes $10 every week and there are 10 people participating, a total of $100 is collected each week. This amount will be distributed among the winners.
In the first draw, you have a 1 in 10 chance of winning, so if you win, you would receive $100.
In the second draw, your chances of winning increase to 1 in 20, so if you win, you would receive $100.
In the third draw, your chances of winning increase to 1 in 30, so if you win, you would receive $100.
This pattern continues until the fifth draw, where your chances of winning are 1 in 50, again resulting in a $100 prize if you win.
To calculate the expected size of the prize, we need to multiply the probability of winning each week by the prize amount and sum them up.
(1/10) * $100 + (1/20) * $100 + (1/30) * $100 + (1/40) * $100 + (1/50) * $100 = $10 + $5 + $3.33 + $2.5 + $2 = $22.83
Therefore, the expected size of the prize you would win is approximately $22.83.
c) Whether this game is fair depends on your definition of fairness. In terms of the probability of winning, each participant has an increasing chance as the draws progress, which could be considered fair. However, the expected size of the prize is not evenly distributed among the participants. The first few winners receive the same $100 prize, while the last winner(s) receive the same $100 prize, resulting in an uneven distribution.
If fairness is defined as equal probabilities and equal expected prize sizes for each participant throughout the game, then this game is not fair.