The Texas Lotto is a twice-weekly lottery where players pick any 6 of 54 possibl

Question

The Texas Lotto is a twice-weekly lottery where players pick any 6 of 54 possible numbers. The state picks its own set of 6 numbers, and prizes are awarded based on the number of matching numbers (all six is worth millions). The probability of picking w winning numbers is the hypergeometric probability Pr(W = w) = (6 w) (48 6 - w)/(5.4 6) (a) Define a big winner as someone who picks all 6 numbers correctly. Calculate the probability that a player is a big winner in a single Lotto: Pb = Pr(W = 6) (b) Define a little winner as someone who picks 4 or more numbers correctly. Calculate the probability that a player is a little winner in a single Lotto. Pw = Pr(W > 3) (c) Your friend Lupe is a Lotto fanatic, who buys a lotto ticket twice a week, every week of the year. What is the probability that he is a little winner at least once in a year? (d) A recent Lotto sold 20,000,000 tickets. What is the probability that there were two or more big winners?

Explanation / Answer

a) big winner P(w=6) = 6C6 * 48C0 / 54C6

= 1/25827165

ways to pick 6 numbers out of 54 is 54 C 6 ways,out of which only 1 number is the winner.

b) P(w>3) = P (W=4) + P(W = 5) + P (W=6)

= 1/54C6 * ( 6C6 * 48C0 + 6C6 * 48C0 + 6C6 * 48C0 )

= 17209 / 25827165

c) He buys twice a week for one year means, he buys for 2*52 i.e. 104 ways

prob of lil winner is answer b)

He buys 104 times that is n =104 and has to be winner for 1 time ie. x = 1

Prob he is a lil winner atleast once a year = P (x>=1) = 1- p(x=0) = 1 - nCx p^x Q^n-x

= 1- (104C0 * (17209/25827165) ^0 * (25809956/25827165)^104)

= 1 - (0.9993) = 0.00067

d) sold 20 million tkts

p = 1/54C6

x - no. of winners

E(X) = np = 20*10^6* 1/54C6 = 0.7744

The no. of winners out of 20 million tkts follow a poisson model.

P (x>=2) = 1- [ P(x=0) + p(x=1) ]

Using this formula, P (X = x) e^-mu * mu ^x/ x factorial

P (x>=2) = 1 - e^-0.7744 (0.7744^0 + 0.7744^1)/ 1

= 0.6430

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