+123 In the SuperLottery, three balls are drawn (at random) from ten white balls numbered from to , and one SuperBall is drawn (at random) from ten red balls numbered from to . When you buy a ticket, you choose three numbers from to and one number from to .
If the numbers on your ticket match at least two of the white balls or match the red SuperBall, then you win a super prize. What is the probability that you win a super prize?
You can win in any of four ways:
1) Getting two of the three winning white balls and the red SuperBall.
Two of the three wining white balls and one out of seven losing white balls:
3C2·7C1 / 10C3 = 3·7/120 = 7/40
Combine this with getting the red SuperBall = 1/10 ---> 7/400
2) Getting two of the three white balls and not getting the red SuperBall:
The 7/40 from above combined with not getting the red SuperBall = 9/10 ---> 63/400
3) Getting all three of the white balls and the red SuperBall:
Three of three winning white balls and none of the seven losing white balls:
3C3·7C0 / 10C3 = 1·1/120 = 1/120
Combine this with getting the red SuperBall = 1/10 ---> 1/200
4) Getting all three of the white balls and not getting the SuperBall:
The 1/120 from above combined with not getting the red SuperBall = 9/10 ---> 3/400
Adding up all the probabilities, we get the answer of 3/16.
+59 I think you mean that the question is:
In the SuperLottery, three balls are drawn (at random, without replacement) from ten white balls numbered from 1 to 10, and one SuperBall is drawn (at random) from ten red balls numbered from 11 to 20. When you buy a ticket, you choose three numbers from 1 to 10 and one number from 11 to 20.
If the numbers on your ticket match at least two of the white balls or match the red SuperBall, then you win a super prize. What is the probability that you win a super prize?
Answer:
We compute the complement: we'll count the number of losing tickets.
To have a losing ticket, you must have at most one correct white ball, and miss the SuperBall.
You miss all 3 white balls if your ticket contains 3 of the 7 white numbers that were not drawn, so there are \(\dbinom{7}{3}=\dfrac{7\cdot6\cdot5}{6}=35\) possibilities.
You hit 1 white ball and miss the others if your ticket contains 1 of the 3 white numbers that were drawn and 2 of the 7 white numbers that were not drawn, so there are \(3\dbinom{7}{3}=\dfrac{3\cdot7\cdot6}{2}=63\) possibilities.
You miss the SuperBall if you have one of the 9 red numbers that were not drawn.
Therefore, there are \((35+63)\cdot9=882\) losing tickets.
Hence, there are \(1200-882=318\) winning tickets, and your probability of winning a super prize is
\(\dfrac{318}{1200}=\boxed{\dfrac{53}{200}}.\)
Note: We can approach this problem using direct counting, but there are a number of cases:
Given the numerous cases here, a complementary counting approach is a faster approach.
