Probability

[Richard B]

In the SuperLottery, three balls are drawn (at random) from ten white balls numbered from 1to 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?

so far I got

• 67/475
• 7/25

but they are both wrong and now I'm truly stumped and I got so frustrated I've been trying all day I don't even remember now how I got the wrong answers . would someone please help.

 

[pka]

In the SuperLottery, three balls are drawn (at random) from ten white balls numbered from 1to 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?

Let A be the event of matching exact two white numbers.
Let B be the event of matching exact three white numbers.
Let C be the event of matching exact one red number.
What \(\mathcal{P}([A\cup B]\cup C)=~?\) Note that A & B are disjoint, but AB and C are not.

 

[Richard B]

I'm not sure what the answer is, I know I need to find the separate probabilities, but I don't know how to do that (especially for b)

 

[pka]

Say that you have chosen three white numbers. What is the probability that exact two of those three match winning numbers?
Then what is the probability that all three of those match winning numbers?

 

[Richard B]

I don't know what the probability is that exactly two of those three-match winning numbers. or the probability that all three match-winning numbers

 

[pka]

I don't know what the probability is that exactly two of those three-match winning numbers. or the probability that all three match-winning numbers

\(\dbinom{10}{3}\)=\(\dfrac{10!}{3!\cdot 7!}=120\) ways to choose 3 from 10 (three white balls).
\(\dbinom{10}{2}=45\)
Now what?

 

[pka]

@Richard, please don't give up. There are three lucky white balls.
That means there are seven unlucky: \(\dbinom{7}{3}=35\) ways for you to choose zero lucky balls.
Now, there are: \(\dbinom{7}{2}\cdot\dbinom{7}{1}=63\) ways for you to choose one lucky ball .
How ways are there for you to choose at most one lucky number?
How ways are there for you to choose at least two lucky numbers?
If there are 98 ways to choose 0 or 1 in out of 120.
What is the probability of choosing at least two?