Alottery has 100 tickerts which are placed in a barrel . Three tickets are drawn at random from the barrel to decide 3 prizes . If john has 3 tickets in the lottery , determine his probability of winning :

A.first prize. B.first and second prize. C.all 3 prizes. D.none of the prizes

C. all 3 prizes

To determine the probability of John winning each prize, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.

Given that there are 100 tickets in the barrel, the total number of possible outcomes is the number of ways to choose 3 tickets out of 100, which is represented by the combination formula:

nCr = n! / (r! * (n-r)!),

where n is the total number of items and r is the number of items chosen.

For the calculations, we can use the following notation:
n = 100 (total number of tickets)
r = 3 (number of tickets drawn)

A. First Prize:
To determine the probability of John winning the first prize, we need to calculate the number of favorable outcomes, which is simply 1 (as there is only one first prize). Thus, the probability of John winning the first prize is:

P(A) = 1 / nCr(100, 3) = 1 / (100! / (3! * (100-3)!)).

B. First and Second Prize:
To determine the probability of John winning both the first and second prizes, we need to calculate the number of favorable outcomes, which is 1 (since there is only one first prize and one second prize). Thus, the probability of John winning both the first and second prizes is:

P(B) = 1 / nCr(100, 3) = 1 / (100! / (3! * (100-3)!)).

C. All Three Prizes:
To determine the probability of John winning all three prizes, we need to calculate the number of favorable outcomes, which is again 1 (since there is only one set of tickets that can win all three). Thus, the probability of John winning all three prizes is:

P(C) = 1 / nCr(100, 3) = 1 / (100! / (3! * (100-3)!)).

D. None of the Prizes:
To determine the probability of John winning none of the prizes, we calculate the number of favorable outcomes as the number of ways to choose 3 tickets out of the remaining 97 tickets (since John has 3 tickets). Thus, the probability of John winning none of the prizes is:

P(D) = nCr(97, 3) / nCr(100, 3) = (97! / (3! * (97-3)!)) / (100! / (3! * (100-3)!)).

Now, you can use the above formulas to calculate the probabilities of each scenario.

To determine the probability of John winning each prize in the lottery, we need to calculate the ratio of the number of favorable outcomes to the number of possible outcomes for each scenario.

Total tickets in the barrel = 100
Total prizes to be drawn = 3

A. Probability of winning the first prize:
John has 3 tickets, so the number of favorable outcomes for him to win the first prize is 3.
Total possible outcomes for the first prize is the total number of tickets in the barrel, which is 100.
Therefore, the probability of John winning the first prize is 3/100.

B. Probability of winning the first and second prize:
To calculate this probability, we need to consider that once a ticket is drawn, it is not put back into the barrel. Therefore, the number of possible outcomes decreases for each prize.
For John to win the first prize, he has 3 favorable outcomes. After one ticket is drawn, there are 99 tickets left in the barrel.
For John to win the second prize, he still has 3 favorable outcomes. However, now there are only 98 tickets left in the barrel.
The probability of him winning both prizes is (3/100) * (3/99).

C. Probability of winning all three prizes:
Similar to the previous scenario, the number of possible outcomes reduces even further.
For John to win the first prize, he has 3 favorable outcomes out of 100.
For John to win the second prize, there would be 2 favorable outcomes left out of 99 tickets in the barrel.
For John to win the third prize, he would have only 1 favorable outcome left out of 98 tickets.
The probability of winning all three prizes is (3/100) * (2/99) * (1/98).

D. Probability of winning none of the prizes:
If John doesn't win any of the prizes, it means he doesn't have any of the winning tickets.
So the number of favorable outcomes for this scenario is 0.
The probability of winning none of the prizes is therefore 0/100, which is 0.

To summarize:
A. Probability of winning the first prize = 3/100
B. Probability of winning the first and second prize = (3/100) * (3/99)
C. Probability of winning all three prizes = (3/100) * (2/99) * (1/98)
D. Probability of winning none of the prizes = 0/100 = 0