Binomial Distribution
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
\(X\sim B(8,0.3)\). Find:
(a)\(P(X=3)\)
(b)\(P(X\leq2)\)
(c)\(E(X)\)
(d)\(\text{Var}(X)\)
(a)
Formula
\(\binom{8}{3}(0.3)^3(0.7)^5=56\times0.027\times0.16807\)
(b)
\(P(0)+P(1)+P(2)\)
\((0.7)^8+8(0.3)(0.7)^7+28(0.3)^2(0.7)^6\approx0.0576+0.198+0.296\)
(c)
Answer
\(np=8\times0.3=2.4\)
(d)
Answer
\(np(1-p)=2.4\times0.7=1.68\)
Question 2
A test has 10 true/false questions. A student guesses at random. Find the probability they get at least 7 correct.
\(X\sim B(10,0.5)\)
\(P(X\geq7)=P(7)+P(8)+P(9)+P(10)\)
By symmetry and formula
\(\binom{10}{7}+\binom{10}{8}+\binom{10}{9}+\binom{10}{10}\) all divided by \(2^{10}=1024\)
Numerator
\(120+45+10+1=176\)
Answer
\(\dfrac{176}{1024}=\dfrac{11}{64}\approx0.172\)
Question 3
A biased coin has \(P(H)=0.6\). It is tossed 12 times. Find the most likely number of heads.
Mode of binomial
Mode \(=\lfloor(n+1)p
floor=\lfloor13\times0.6
floor=\lfloor7.8
floor=7\)
Question 4
If \(X\sim B(n,0.4)\) and \(E(X)=6\), find \(n\) and \(\text{Var}(X)\).
Solve for \(n\)
\(0.4n=6\Rightarrow n=15\)
Answer
\(n=15\); \(\text{Var}(X)=15\times0.4\times0.6=3.6\)
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