The binomial distribution

About this lesson

Statistics (AP-aligned) › Random variables

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The binomial distribution

Warm-up

Three quick questions from earlier lessons.

Discover

A coin is tossed 3 times. How many sequences have exactly one head?

  1. 3
  2. 1

Key idea

Count the successes in a fixed number of independent trials.

Binomial setting: nn fixed trials, two outcomes, independent, same probability pp of success. Then P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k}p^k(1 - p)^{n - k}, where (nk)\binom{n}{k} counts the arrangements: (n1)=n\binom{n}{1} = n, (n2)=n(n−1)2\binom{n}{2} = \tfrac{n(n - 1)}{2}.

Worked example

n=5n = 5, p=0.2p = 0.2: P(X=2)P(X = 2).

(52)=10\binom{5}{2} = 10
Arrangements.
0.22×0.83=0.020480.2^2 \times 0.8^3 = 0.02048
One arrangement.
P(X=2)=0.2048P(X = 2) = 0.2048
Multiply.

Your turn, guided

Your turn: n=4n = 4, p=12p = \tfrac{1}{2}. P(X=1)P(X = 1).

14\tfrac{1}{4}
4 × (½)⁴.

Practice ★

  1. A free throw is made with probability 1/6. In 5 independent shots, find P(exactly one is made).
  2. A free throw is made with probability 1/5. In 6 independent shots, find P(exactly one is made).
  1. Which situation fits a binomial model?
    1. The number of heads in 9 tosses of a fair coin
    2. The number of tosses until the first head
    3. The number of aces in 9 cards dealt from one deck

Watch out!

Practice ★★

  1. A seed sprouts with probability 1/4, independently. 6 seeds are planted. Find P(exactly two sprout).
  2. A seed sprouts with probability 1/3, independently. 8 seeds are planted. Find P(exactly two sprout).
  1. Boss problem. A machine part is faulty with probability 1/5, independently. Four parts are tested; X counts the faulty ones.

Exit ticket

Three questions to check you have got it.

  1. A free throw is made with probability 1/6. In 5 independent shots, find P(exactly one is made).
  2. A free throw is made with probability 1/5. In 6 independent shots, find P(exactly one is made).
  3. A free throw is made with probability 1/2. In 7 independent shots, find P(exactly one is made).

Summary card

Binomial probabilities

P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k}p^k(1 - p)^{n - k} in a binomial setting.

Watch out: multiply by the number of arrangements.

End of the lesson

Next lesson: Mean and standard deviation of a binomial variable Back to the chapter

Answers: The binomial distribution

A

  1. 31257776\frac{3125}{7776}
  2. 614415625\frac{6144}{15625}

B

  1. The number of heads in 9 tosses of a fair coin

C

  1. 12154096\frac{1215}{4096}
  2. 17926561\frac{1792}{6561}

D

  1. 45\frac{4}{5}

E

  1. 31257776\frac{3125}{7776}
  2. 614415625\frac{6144}{15625}
  3. 7128\frac{7}{128}

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