Answer in Statistics and Probability for Robin #118622
Q1.Two cards are drawn successively without replacement from a wellshuffled deck of 52 cards. Find the probability distribution of the number of aces.
Let X denote the number of aces in a sample of 2 cards drawn from a wellshuffled deck of 52 cards. Then X can take the values 0, 1 and 2.
“P(X=0)=P(no ace)={48 over 52}cdot{47 over 51}={188 over 221}”
“P(X=1)=P(one ace)={4 over 52}cdot{48 over 51}+{48 over 52}cdot{4 over 51}={32 over 221}”
“P(X=2)=P(two aces)={4 over 52}cdot{3 over 51}={1 over 221}”
The probability distribution of X is given by
“begin{matrix}n X=x & 0 & 1 & 2 \n\n p(X=x) & dfrac{188}{221} & dfrac{32}{221} & dfrac{1}{221}nend{matrix}”
Q2. Let X denote the number of defective bolts in a sample of n bolts: “Xsim Bin(n,p)”
“P(X=x)=binom{n}{k}p^x(1-p)^{n-x}”
Given “p=0.2,n=5.”
“P(X=1)+P(X=2)=”
“=binom{5}{1}(0.2)^1(1-0.2)^{5-1}+binom{5}{2}(0.2)^2(1-0.2)^{5-2}=”
“=5(0.2)(0.8)^4+10(0.2)^2(0.8)^3=0.6144”
Probability that a second sample is required “=0.6144.”
