Chapter 8.6, Problem 23E

To determine

To find: the probability of winning a lottery that a player who buys one ticket matches all n numbers drawn (in any order ).

Answer to Problem 23E

Probability of winning is $P(E)=\frac{1}{38,38,380}.$

Explanation of Solution

Given information: A player chooses n numbers from a to b in a lottery. Where n =6,

a =1, b =40.

Formula used:

If an event has n(E) equally likely outcomes and its sample space S has n(S) equally likely outcomes, then the probability of event is given by,

  $P(E)=\frac{n(E)}{n(S)}.$

Formula for the number of combinations of n numbers taken r numbers at a time is,

  $n(S){=}_n{C}_r.$

Calculation:

To find the outcomes in the sample space, use the formula for the number of combination of 40 numbers taken 6 numbers at a time.

  $n(S){=}_n{C}_r{=}_{40}{C}_6=\frac{40\times 39\times 38\times 37\times 36\times 35}{6\times 5\times 4\times 3\times 2\times 1}=38,38,380.$

Player buys one ticket, n ( E) =1 ,the probability of winning is,

  $P(E)=\frac{n(E)}{n(S)}=\frac{1}{38,38,380}.$