The UK national lottery works as follows. You choose six (different) numbers in the range 1 to 49. If all six come up in the draw (in any order) you win the first prize, expected to be around £2m (which could be shared if someone else chooses the six winning numbers).

(a) What is your chance of winning with a single ticket?

(b) You win a second prize if you get five out of six right, and your final chosen number matches the ‘bonus’ number in the draw (also in the range 1 to 49). What is the probability of winning a second prize?

(c) Calculate the probabilities of winning a third, fourth or fifth prize, where a third prize is won by matching five out of the six numbers, a fourth prize by matching four out of six and a fifth prize by matching three out of six.

(d) What is the probability of winning a prize? 

(e) The prizes are as follows:

Comment upon the distribution of the fund between first, second, etc. prizes.

(f) Why is the fifth prize guaranteed whereas the others are not?

(g) In the first week of the lottery, 49 million tickets were sold. There were 1 150 000 winners, of which 7 won (a share of) the jackpot, 39 won a second prize, 2139 won a third prize and 76 731 a fourth prize. Are you surprised by these results or are they as you would expect?

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Prize Value £2 million £100 000 First (expected, possibly shared) (expected, for each winner) (expected, for each winner) (expected, for each winner) (guaranteed, for each winner) Second Third £1500 Fourth £65 Fifth £10