Answer:
(a) 0.0024
(b) 0.0012
(c) 0.0006
(d) 0.0004
Step-by-step explanation:
The total number of possible integers when any number is selected is 10 (i.e from 0 - 9). When four number integers are selected, the total number of sample sample will be;
10 × 10 × 10 × 10 = 10,000
The sample space = 10,000
To know the possible ways of selecting the given four digits, we will use permutation.
[tex]^{n}P_{r} = \frac{n!}{(n-r)!}[/tex]
To get the probability,
[tex]Probability \ of \ winning (Selected \ numbers) = \frac{number\ of\ possible\ outcomes\ of\ selected\ numbers}{sample\ space}[/tex]
(a) When 6,7,8,9 are selected, n = 4 , r = 4
The possible ways of selecting 6,7,8,9 is;
[tex]^{4}P_{4} = \frac{4!}{(4-4)!}[/tex]
[tex]= \frac{4!}{(0)!}[/tex]
but 0! = 1
[tex]^{4}P_{4} = 4![/tex]
= 4 × 3 × 2 × 1 = 24
[tex]Prob (6,7,8,9) = \frac{24}{10000} = 0.0024[/tex]
(b) When 6, 7, 8, 8 are selected,
The possible ways of selecting 6,7,8,8 is;
[tex]= \frac{4!}{1! \ 1! \ 2!}[/tex]
[tex]= \frac{4!}{2!}[/tex]
[tex]=\frac{4 * 3 * 2 * 1}{2 * 1}[/tex]
= 12
[tex]Prob (6,7,8,8) = \frac{12}{10000} = 0.0012[/tex]
(c) When 7, 7, 8, 8 are selected,
The possible ways of selecting 7,7,8,8 is;
[tex]= \frac{4!}{2! \ 2!}[/tex]
[tex]=\frac{4 * 3 * 2 * 1}{(2 * 1)(2 * 1)}[/tex]
= 6
[tex]Prob (7,7,8,8) = \frac{6}{10000} = 0.0006[/tex]
(d) When 7, 8, 8, 8 are selected,
The possible ways of selecting 7,8,8,8 is;
[tex]= \frac{4!}{1! \ 3!}[/tex]
[tex]=\frac{4 * 3 * 2 * 1}{3 * 2 * 1}[/tex]
= 4
[tex]Prob (7,8,8,8) = \frac{4}{10000} = 0.0004[/tex]
