Calculate the odds of winning the Mega Millions prize of $4. In order to win, yo
ID: 3049293 • Letter: C
Question
Calculate the odds of winning the Mega Millions prize of $4. In order to win, you must match 1 (and only 1) of the winning 5 white balls numbered 1 to 70, and match the yellow Mega ball numbered 1 to 25. Write your final answer in 1/p instead of the fraction p. For example, ¼ = 4. 0.10 = 10. Note: There are 2 separate bags – one with the 70 white balls and another with the 25 yellow balls. For the white balls, we are picking without replacement. (Note: If necessary, draw a bag with 70 balls in it and mark 5 as the winning balls. As you draw each ball out of the bag, cross out that ball in your bag.)
What is the probability of picking 1 of the 5 winning white balls from the bag? 5/70
What is the probability of NOT picking 1 of the remaining 4 winning white balls from the bag for your 2nd pick? 4/69
What is the probability of NOT picking 1 of the remaining 4 winning white balls from the bag for your 3rd pick?
What is the probability of NOT picking 1 of the remaining 4 winning white balls from the bag for your 4th pick?
What is the probability of NOT picking 1 of the remaining 4 winning white ball from the bag for your 5th pick?
What is the probability of picking the winning yellow Mega ball from the other bag?
Now we picked the white balls in a certain order : WLLLL, one of the winning balls first, and then 4 losing balls. But the order really does not matter. We could have picked LWLLL, LLWLL, etc. We have to factor this into our probability. So what number do we have to multiply our probability to account that the order of picking does not matter?
Using the multiplication rule for the numbers you got for a-g, what is the probability of winning the Mega Millions $4 prize in 1/p? (Note: The Mega Millions ticket says the probability of winning are 1 out of 89. If you do not get something close to this number, then you did something wrong!)
Explanation / Answer
The probability of picking 1 of the 5 winning white balls from the bag = C[5,1] / C[70,1] = 5/70
The probability of NOT picking 1 of the remaining 4 winning white balls from the bag for the 2nd pick = 1- 4/69 = 65/69
The probability of NOT picking 1 of the remaining 4 winning white balls from the bag for the 3rd pick = 64/68
The probability of NOT picking 1 of the remaining 4 winning white balls from the bag for the 4th pick = 63/67
The probability of NOT picking 1 of the remaining 4 winning white ball from the bag for the 5th pick = 62/66
The probability of picking the winning yellow Mega ball from the other bag = 1/25
To account for the order of picking the winning ball from the 5 specified balls, there are C[5,1] = 5 ways
So the probability of winning the Mega Millions $4 prize = (5/70) x (65/69) x (64/68) x (63/67) x (62/66) x (1/25) x 5 = 1/89.4
Expressed in 1/p form, the odds is 89.4
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