Suppose in a certain (large) community, 40% of the population is under 30 years

Question

Suppose in a certain (large) community, 40% of the population is under 30 years of age, 30% of the population is between 30 and 50 years of age, 20% of the population is between 50 and 70 years of age, and 10%. of the population is over 70. If eight people are randomly selected from the community, find the probability that (a) exactly four are under 30, exactly two are between 30 and 50, exactly one is between 50 and 70, and exactly one is over 70. (b) exactly three are under 30 and exactly two are between 50 and 70. (c) exactly five are under 50.

Explanation / Answer

a) The probability of succes (to be within th epopulation of under 30) is 0.4, total number of trials, n=8, r=4 denotes specific number of success in n trials. Use, P(X,r)=nCr(p)^r(q)^n-r

P(X=4)=8C4(0.4)^4(0.6)^4=0.2322

P(X=2)=8C2(0.3)^2(0.7)^6=0.296

P(X=1)=8C1(0.2)^1(0.8)^7=0.336

P(X=1)=8C1(0.1)^1(0.9)^7=0.383

Required probability is:0.383*0.336*0.296*0.232=0.008

b) P(X=3)=8C3(0.4)^3(0.6)^5=0.279

P(X=2)=8C2(0.2)^2(0.8)^6=0.294

Required probability is: 0.279*0.294=0.082

c) P(X=5)=8C5(0.3)^5(0.7)^3=0.0467

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