In Shatang-Land\'s lottery, players pick seven different integers between 1 and

Question

In Shatang-Land's lottery, players pick seven different integers between 1 and 54, order of selection is irrelevant. Seven numbers among -54 are randomly selected as winning numbers. A player hits the jackpot if she/he matches all seven numbers. The second big prize is rewarded to person(s) matching six numbers and the third prize goes to person(s) matching 5. CCCXhi is a player. Find the probabilities that: CCCXhi's ticket wins the jackpot. CCCXhi's ticket wins the second prize. CCCXhi's ticket wins the third prize.

Explanation / Answer

Step 1: Define terms and variables.

Let X = # of winning numbers. Possible values are: 0, 1, 2,3,…, 7

X ~ Hypergeometric(G = 7, n = 7, N = 54)

G, n, and N are defined below.

G = 7, which is the number of wining numbers.

B = 47, which is the number non-wining numbers.

N = 57, which is the total number.

n = 7, which is the sample size.

nCx = n! / [x! * (n–x)!]

P(X=x) = GCx * BC(n– x) / NCn

Step 2: Find the probability distribution.

P(X = x) = 7Cx *47C(7 – x) /54C7

P(X = 7) = 7C7 *47C0 / 54C7=5.647*10-9

P(X = 6) = 7C6 *47C1 / 54C7=1.858*10-6

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