The chance of selecting 10 men is unlikely in a random selection
The probability that the jury is made of 2 men
The given parameters are:
- Men = 17
- Women = 15
- Total = 32
The total number of selections is:
[tex]Total = ^{32}C_{10}[/tex]
Evaluate
Total = 64512240
If 2 men are selected, then 8 women would be selected.
So, the number of ways is:
[tex]Ways = ^{17}C_2 * ^{15}C_8[/tex]
Evaluate
Ways = 136 * 6435
Ways = 875160
The required probability is:
P = 875160/64512240
P = 0.0136
Hence, the probability that the jury is made of 2 men is 0.0136
The probability that half the jury are men
If 5 men are selected, then 5 women would be selected.
So, the number of ways is:
[tex]Ways = ^{17}C_5 * ^{15}C_5[/tex]
Evaluate
Ways = 6188 * 792
Ways = 4900896
The required probability is:
P = 4900896/64512240
P = 0.0760
Hence, the probability that the jury is made of half men and women is 0.0760
The probability that at least two women are selected
To do this, we make use of the complement rule:
P = 1 - P(Not up to 2 is selected)
When 0 women are selected, then 10 men are selected and when 1 woman is selected, then 9 men are selected
So, the number of ways is:
[tex]Ways = ^{17}C_{10} * ^{15}C_0 + ^{17}C_{9} * ^{15}C_1[/tex]
Evaluate
Ways = 19448 * 1 + 24310 * 15
Ways = 384098
The probability P(Not up to 2 is selected) is:
P = 0.0060
The required probability is:
P = 1 - 0.0060
P = 0.9940
Hence, the probability that the jury is made of at least two women is 0.9940
The probability that at 10 men are selected
When 10 men are selected, then 0 women are selected
So, the number of ways is:
[tex]Ways = ^{17}C_{10} * ^{15}C_0[/tex]
Evaluate
Ways = 19448 * 1
Ways = 19448
The probability is:
P = 19448/64512240
P = 0.0003
Hence, the probability that the jury is made of 10 men is 0.0003
The selection above can lead to suspicion because the chance of selecting 10 men is unlikely in a random selection
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