Binomial Distribution — Practice Problems and Worked Solutions

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17 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find P(X=k) for the binomial random variable.

   $$X\sim \mathrm{Bin}(4,\frac{1}{2}),k=2$$

   Answer: $\frac{3}{8}$

   Step-by-step solution

   1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
   2. $$=\frac{3}{8}$$
2. 2.

   Find P(X=k) for the binomial random variable.

   $$X\sim \mathrm{Bin}(5,\frac{1}{2}),k=3$$

   Answer: $\frac{5}{16}$

   Step-by-step solution

   1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
   2. $$=\frac{5}{16}$$
3. 3.

   Find P(X=k) for the binomial random variable.

   $$X\sim \mathrm{Bin}(3,\frac{1}{3}),k=1$$

   Answer: $\frac{4}{9}$

   Step-by-step solution

   1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
   2. $$=\frac{4}{9}$$
4. 4.

   Find P(X=k) for the binomial random variable.

   $$X\sim \mathrm{Bin}(6,\frac{1}{2}),k=0$$

   Answer: $\frac{1}{64}$

   Step-by-step solution

   1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
   2. $$=\frac{1}{64}$$
5. 5.

   Find P(X=k) for the binomial random variable.

   $$X\sim \mathrm{Bin}(5,\frac{1}{5}),k=2$$

   Answer: $\frac{128}{625}$

   Step-by-step solution

   1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
   2. $$=\frac{128}{625}$$
6. 6.

   Find P(X=k) for the binomial random variable.

   $$X\sim \mathrm{Bin}(8,\frac{3}{4}),k=6$$

   Answer: $\frac{5103}{16384}$

   Step-by-step solution

   1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
   2. $$=\frac{5103}{16384}$$
7. 7.

   Find P(X=k) for the binomial random variable.

   $$X\sim \mathrm{Bin}(4,\frac{1}{3}),k=2$$

   Answer: $\frac{8}{27}$

   Step-by-step solution

   1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
   2. $$=\frac{8}{27}$$
8. 8.

   Find P(X=k) for the binomial random variable.

   $$X\sim \mathrm{Bin}(6,\frac{1}{3}),k=2$$

   Answer: $\frac{80}{243}$

   Step-by-step solution

   1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
   2. $$=\frac{80}{243}$$
9. 9.

   Find P(X=k) for the binomial random variable.

   $$X\sim \mathrm{Bin}(5,\frac{2}{5}),k=3$$

   Answer: $\frac{144}{625}$

   Step-by-step solution

   1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
   2. $$=\frac{144}{625}$$
10. 10.

    Find P(X=k) for the binomial random variable.

    $$X\sim \mathrm{Bin}(7,\frac{1}{2}),k=4$$

    Answer: $\frac{35}{128}$

    Step-by-step solution

    1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
    2. $$=\frac{35}{128}$$
11. 11.

    Find P(X=k) for the binomial random variable.

    $$X\sim \mathrm{Bin}(3,\frac{1}{2}),k=2$$

    Answer: $\frac{3}{8}$

    Step-by-step solution

    1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
    2. $$=\frac{3}{8}$$
12. 12.

    Find P(X=k) for the binomial random variable.

    $$X\sim \mathrm{Bin}(8,\frac{1}{4}),k=2$$

    Answer: $\frac{5103}{16384}$

    Step-by-step solution

    1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
    2. $$=\frac{5103}{16384}$$
13. 13.

    Find P(X=k) for the binomial random variable.

    $$X\sim \mathrm{Bin}(6,\frac{2}{3}),k=4$$

    Answer: $\frac{80}{243}$

    Step-by-step solution

    1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
    2. $$=\frac{80}{243}$$
14. 14.

    Find P(X=k) for the binomial random variable.

    $$X\sim \mathrm{Bin}(4,\frac{3}{4}),k=3$$

    Answer: $\frac{27}{64}$

    Step-by-step solution

    1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
    2. $$=\frac{27}{64}$$
15. 15.

    Find P(X=k) for the binomial random variable.

    $$X\sim \mathrm{Bin}(10,\frac{1}{2}),k=5$$

    Answer: $\frac{63}{256}$

    Step-by-step solution

    1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
    2. $$=\frac{63}{256}$$
16. 16.

    Find P(X=k) for the binomial random variable.

    $$X\sim \mathrm{Bin}(5,\frac{3}{5}),k=3$$

    Answer: $\frac{216}{625}$

    Step-by-step solution

    1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
    2. $$=\frac{216}{625}$$
17. 17.

    Find P(X=k) for the binomial random variable.

    $$X\sim \mathrm{Bin}(9,\frac{1}{3}),k=3$$

    Answer: $\frac{1792}{6561}$

    Step-by-step solution

    1. $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$
    2. $$=\frac{1792}{6561}$$

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