The prize winners for a lottery are determined by drawing 5 balls from a drum of
ID: 3241692 • Letter: T
Question
The prize winners for a lottery are determined by drawing 5 balls from a drum of 53 white numbered balls and 1 ball from a drum of 42 black numbered balls. A lottery ticket that matches the numbers of all 6 balls wins the jackpot. A ticket that matches only the number of the black ball wins $3.
(a) Explain how to use counting techniques to find the probability of winning the jackpot. Include the probability of winning the jackpot.
(b) The lottery organizers state that the probability of matching only the black ball is 1/70. One of your students says it should be 1/42.
Explain what your student is missing. (Keep in mind what has to happen to match only the black ball.) Then show how to use the Fundamental Counting Principle to find the probability.Explanation / Answer
a) 5 balls need to be selected from 53 white balls and 1 ball from 42 black numbered balls.
Using combinations, this can be done in 53C5 * 42C1 = 120526770 ways
The number of ways your first 5 numbers can match the 5 White numbers is 5C5 = 1
The number of ways your black number can match the drawn black number is 1C1 = 1
The probability of jackpot is [5C5 * 1C1]/[53C5*42C1] = 1/120526770
b) The student is considering only the number of ways to match your black number with the drawn black number, but fails to consider the number of ways to not match the 5 white numbers.
The number of ways your black number can match the drawn black number is 1C1 = 1
The number of ways your first 5 numbers does not match the 5 White numbers is 5C0 = 1
The number of ways your 5 numbers can match the remaining 48 white numbered balls = 48C5
Therefore probability of matching only the black number = [5C0*48C5*1C1]/[53C5 * 42C1] = 0.0142 which is approximately 1/70
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