Question

Assume that a winning ticket is one which matches the 6 numbers drawn from 1 to 49.
(a) (1 mark) Suppose p is the probability of winning the grand prize. Write down the value for p for
Lotto 649.
(b) (1 mark) Write down the probability of winning (for the first time) on the nth draw (i.e. losing
on the first n − 1 draws).
(c) (1 mark) Determine the expected number of draws you must play (1 ticket each draw) before
winning for the first time.
(d) (1 mark) Show how the average time to win Lotto 649 when playing 1 ticket per weekly 649 draw
turns into the long wait given for the Homo sapiens example.

Answer 1

Here we assume that a winning ticket is one which matches the 6 numbers drawn from 1 to 49.

(a) (1 mark) Suppose p is the probability of winning the grand prize. Write down the value for p for
Lotto 649.

The total ways of drawing 6 numbers from the 49 numbers are ⁴⁹C₆ .

Let's use excel:

⁴⁹C₆ = "=COMBIN(49,6)" = 13983816

The possible ways of selecting 6 numbers which are match the winning numbers on the ticket is

⁶C₆ = 1

So required probability = p = 1/13983816 = 0.000000715112.

(b) (1 mark) Write down the probability of winning (for the first time) on the nth draw (i.e. losing
on the first n − 1 draws).

Let p = probability of winning.

therefore , 1 - p = probability of lossing.

Therefore required probability = (1 - p)^{n - 1} * p (where = p = 0.000000715112

Let X = number of ticket required to first win. So X takes following values.

X = { 1, 2 , 3 .....)

So X follows geometric distribution with probability of success = p = 0.000000715112.

The formula of expected value of geometric distribution is 1/p

Therefore E(X) = 13983816.

(d) (1 mark) Show how the average time to win Lotto 649 when playing 1 ticket per weekly 649 draw
turns into the long wait given for the Homo sapiens example.

The average time to win Lotto 649 when playing 1 ticket is

7*13983816 = 97886712 days = 97886712 /365 = 268182.77 years.