A school is running a raffle. There are 100 tickets, of which 3 are winners. You can assume that tickets are sold by drawing at random without replacement from the available tickets. Teacher X buys 10 raffle tickets, and so does Teacher Y. Find the chance that one of those two teachers gets all three winning tickets.
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To find the chance that one of the two teachers, X or Y, gets all three winning tickets, we need to calculate the probability for each teacher individually and then add them together.
Let's start with Teacher X. We know that Teacher X bought 10 tickets, but we don't know which specific tickets they are. The probability of Teacher X getting all three winning tickets is the same as the probability of selecting all three winning tickets among the 10 tickets that Teacher X bought.
To calculate this probability, we need to consider the total number of ways to select 3 winning tickets out of 100 and divide it by the total number of ways to select 3 tickets out of the 10 that Teacher X bought.
The total number of ways to select 3 winning tickets out of 100 is given by the combination formula, which is nCr (n choose r) and can be calculated as:
nCr = n! / (r!(n-r)!)
In this case, n (total number of tickets) is 100 and r (number of winning tickets) is 3. So, the number of ways to select 3 winning tickets out of 100 is:
100C3 = 100! / (3!(100-3)!) = 100! / (3!97!) = (100 * 99 * 98) / (3 * 2 * 1) = 161,700
Now, we need to calculate the total number of ways to select 3 tickets out of the 10 that Teacher X bought:
10C3 = 10! / (3!(10-3)!) = 10! / (3!7!) = (10 * 9 * 8) / (3 * 2 * 1) = 120
So, the probability of Teacher X getting all three winning tickets is:
P(X) = 120 / 161,700
Similarly, we can calculate the probability for Teacher Y:
P(Y) = 120 / 161,700
To find the chance that one of the two teachers gets all three winning tickets, we add their individual probabilities together:
P(X or Y) = P(X) + P(Y) = 120 / 161,700 + 120 / 161,700
P(X or Y) = 240 / 161,700
Therefore, the chance that one of the two teachers, X or Y, gets all three winning tickets is 240 / 161,700.