# Probability Question

• Oct 11th 2011, 07:40 PM
KelvinScale
Probability Question
Hi everyone,

I am having some trouble getting this question, because it is not a style I have done before.

In a large population, people are one of three genetic types: A, B, C: 30% are type A, 60% Type B, and 10% Type C. The probability a person carries another gene miking them susceptible for a disease is 0.05 for A, 0.04 for B, and 0.02 for C.

If ten unrelated persons are selected, what is the probability at least one is susceptible for the disease?

The answer is 0.342
• Oct 11th 2011, 08:40 PM
Soroban
Re: Probability Question
Hello, KelvinScale!

Quote:

In a large population, people are one of three genetic types: A, B, C:
. . 30% are Type A, 60% are Type B, and 10% are Type C.

The probability a person carries certain gene making them susceptible for a disease
. . is 0.05 for A, 0.04 for B, and 0.02 for C.

If ten people are selected randomly, what is the probability at least one has that gene?

30% of the population are Type A; 5% of them have the gene.
. . We have: .$\displaystyle 0.30 \times 0.05 \:=\:0.015$
1.5% of the population are Type A and have the gene.

60% of the population are Type B; 4% of them have the gene.
. . We have: .$\displaystyle 0.60 \times 0.04 \:=\:0.024$
2.4% of the population are Type B and have the gene.

10% of the population are type C; 2% of them have the gene.
. . We have: .$\displaystyle 0.10 \times 0.02 \:=\:0.002$
0.2% of population are Type C and have the gene.

Hence: .$\displaystyle 1.5\% + 2.4\% + 0.2\% \:=\:4.1\%$ of the population has the gene.
. . and: .$\displaystyle 95.9\%$ do not have the gene.

Ten people are selected at random.
We want probability that at least one of them has the gene.

The opposite of "at least one has the gene" is "none of them has the gene".

We know that:.$\displaystyle P(\text{none has the gene}) \:=\: (0.959)^{10} \:=\:0.657939669 \:\approx\:0.658$

Therefore: .$\displaystyle P(\text{at least one has the gene}) \:=\:1 - 0.658 \;=\;0.342$

• Oct 12th 2011, 02:20 PM
KelvinScale
Re: Probability Question
Thanks!, I knew it was not as hard as I thought.