A-Level The binomial distribution Practice Questions

Free A-Level The binomial distribution practice questions with full step-by-step worked solutions. Covers binomial distribution, B(n,p), P(X=r), mean np. Practise exam-style problems and check your method.

binomial distributionB(n,p)P(X=r)mean npconditionsidentify n and p
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
A fair coin is tossed 66 times. Let XX be the number of heads, so X∼B(6,0.5)X\sim B(6,0.5). Find P(X=3)P(X=3), giving your answer to 4 significant figures.
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Worked solution

  1. Write the distribution

    X∼B(6, 0.5)X\sim B(6,\ 0.5)

    X follows a binomial distribution.

  2. Substitute n, p and r

    P(X=3)=(63)(0.5)3(0.5)3P(X=3)=\binom{6}{3}(0.5)^{3}(0.5)^{3}

    Substitute the known values into the formula.

  3. Compute the probability

    P(X=3)=0.3125P(X=3)=0.3125

    Evaluate the product.

Answer
P(X=3)=0.3125P(X=3)=0.3125
Question 2
2 markseasy
30%30\% of adults in a town own a bicycle. A random sample of 5050 adults is chosen and XX is the number who own a bicycle. Which is the correct distribution of XX?
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Worked solution

  1. Identify the number of trials

    n=number of trialsn=\text{number of trials}

    The number of repetitions is n.

  2. Identify the probability of success

    p=P(success)p=P(\text{success})

    The success probability is p.

  3. State the model

    X∼B(50,0.3)X\sim B(50,0.3)

    n = 50 and p = 0.3.

Answer
X∼B(50,0.3)X\sim B(50,0.3)
Question 3
3 marksintermediate
For X∼B(n,p)X\sim B(n,p), which expression equals the probability of at least 22 successes?
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Worked solution

  1. Translate the words into symbols

    words→notation\text{words}\to\text{notation}

    Rewrite the description in probability notation.

  2. Recall cumulative notation

    P(X≤r)P(X\le r)

    Cumulative tables give P(X<=r).

  3. Use the complement for 'at least'

    P(X≥r)=1−P(X≤r−1)P(X\ge r)=1-P(X\le r-1)

    At least r is one minus P(X<=r-1).

  4. Use the complement for 'more than'

    P(X>r)=1−P(X≤r)P(X>r)=1-P(X\le r)

    More than r is one minus P(X<=r).

  5. Handle 'fewer than'

    P(X<r)=P(X≤r−1)P(X<r)=P(X\le r-1)

    Fewer than r means at most r-1.

  6. State the equivalent expression

    P(X≥2)=1−P(X≤1)P(X\ge 2)=1-P(X\le 1)

    At least 2 is the complement of at most 1.

Answer
1−P(X≤1)1-P(X\le 1)
Question 4
5 markshard
For X∼B(n,p)X\sim B(n,p), which expression equals P(4≤X≤9)P(4\le X\le 9) using cumulative probabilities?
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Worked solution

  1. Translate the words into symbols

    words→notation\text{words}\to\text{notation}

    Rewrite the description in probability notation.

  2. Recall cumulative notation

    P(X≤r)P(X\le r)

    Cumulative tables give P(X<=r).

  3. Use the complement for 'at least'

    P(X≥r)=1−P(X≤r−1)P(X\ge r)=1-P(X\le r-1)

    At least r is one minus P(X<=r-1).

  4. Use the complement for 'more than'

    P(X>r)=1−P(X≤r)P(X>r)=1-P(X\le r)

    More than r is one minus P(X<=r).

  5. Handle 'fewer than'

    P(X<r)=P(X≤r−1)P(X<r)=P(X\le r-1)

    Fewer than r means at most r-1.

  6. Handle a range of values

    P(a≤X≤b)=P(X≤b)−P(X≤a−1)P(a\le X\le b)=P(X\le b)-P(X\le a-1)

    A range is a difference of cumulative probabilities.

  7. Distinguish strict and inclusive inequalities

    < versus ≤<\ \text{versus}\ \le

    Be careful with strict versus inclusive inequalities.

  8. Express 'exactly'

    P(X=r)P(X=r)

    Exactly r successes is a single term.

  9. Convert everything to cumulative form

    tables give P(X≤r)\text{tables give }P(X\le r)

    Rewrite each probability using cumulatives.

  10. State the equivalent expression

    P(4≤X≤9)=P(X≤9)−P(X≤3)P(4\le X\le 9)=P(X\le 9)-P(X\le 3)

    Subtract the cumulative probability up to one below the lower limit.

Answer
P(X≤9)−P(X≤3)P(X\le 9)-P(X\le 3)
Question 5
8 markschallenging
For X∼B(n,p)X\sim B(n,p), which expression equals P(X>5)P(X>5), the probability of more than 55 successes?
Show worked solution

Worked solution

  1. Translate the words into symbols

    words→notation\text{words}\to\text{notation}

    Rewrite the description in probability notation.

  2. Recall cumulative notation

    P(X≤r)P(X\le r)

    Cumulative tables give P(X<=r).

  3. Use the complement for 'at least'

    P(X≥r)=1−P(X≤r−1)P(X\ge r)=1-P(X\le r-1)

    At least r is one minus P(X<=r-1).

  4. Use the complement for 'more than'

    P(X>r)=1−P(X≤r)P(X>r)=1-P(X\le r)

    More than r is one minus P(X<=r).

  5. Handle 'fewer than'

    P(X<r)=P(X≤r−1)P(X<r)=P(X\le r-1)

    Fewer than r means at most r-1.

  6. Handle a range of values

    P(a≤X≤b)=P(X≤b)−P(X≤a−1)P(a\le X\le b)=P(X\le b)-P(X\le a-1)

    A range is a difference of cumulative probabilities.

  7. Distinguish strict and inclusive inequalities

    < versus ≤<\ \text{versus}\ \le

    Be careful with strict versus inclusive inequalities.

  8. Express 'exactly'

    P(X=r)P(X=r)

    Exactly r successes is a single term.

  9. Convert everything to cumulative form

    tables give P(X≤r)\text{tables give }P(X\le r)

    Rewrite each probability using cumulatives.

  10. Check the boundary values

    r−1, r, r+1r-1,\ r,\ r+1

    Check whether the endpoints are included.

  11. Avoid off-by-one errors

    P(X≥r)≠1−P(X≤r)P(X\ge r)\ne 1-P(X\le r)

    A common slip is forgetting the r-1.

  12. Rewrite the target expression

    1−P(X≤r−1)1-P(X\le r-1)

    Rewrite the target using cumulative probabilities.

  13. Compare with the options

    match the expression\text{match the expression}

    Compare the derived expression with the options.

  14. Select the matching statement

    ⇒choose\Rightarrow\text{choose}

    Choose the equivalent expression.

  15. State the equivalent expression

    P(X>5)=1−P(X≤5)P(X>5)=1-P(X\le 5)

    More than 5 excludes 5, so subtract P(X<=5) from 1.

Answer
1−P(X≤5)1-P(X\le 5)

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