Hard A-Level The binomial distribution Questions

Challenging, exam-style A-Level The binomial distribution questions with worked solutions. Stretch yourself on the hardest binomial distribution, B(n,p), P(X=r), cumulative problems.

binomial distributionB(n,p)P(X=r)cumulativeP(X<=r)P(X>=r)
A-Level34 questionsStep-by-step solutions
Question 1
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?
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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. 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)
Question 2
8 markschallenging
A random sample of 6060 voters is taken from a large electorate where 42%42\% intend to vote for a party. Let XX be the number in the sample who do not intend to vote for the party. Which model is correct?
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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. Define what counts as a success

    success=target outcome\text{success}=\text{target outcome}

    Decide which outcome counts as a success.

  4. Recall the parameter order

    X∼B(n,p)X\sim B(n,p)

    Trials n come first, then probability p.

  5. Express the probability as a decimal

    0<p<10<p<1

    p must be a decimal strictly between 0 and 1.

  6. Reject reversed-parameter options

    B(p,n) is wrongB(p,n)\text{ is wrong}

    B(p,n) has the parameters the wrong way round.

  7. Use success rather than failure probability

    p≠1−pp\ne 1-p

    Use the probability of success, not of failure.

  8. Match n to the sample size

    n=sample sizen=\text{sample size}

    The sample size is the number of trials.

  9. Assemble the parameters

    B(n, p) formB(n,\,p)\text{ form}

    Combine n and p in the correct order.

  10. Reject failure-probability options

    1−p models failure1-p\text{ models failure}

    Options using 1-p model failures, not successes.

  11. Reject parameter-swap options

    n↔p swapn\leftrightarrow p\text{ swap}

    Do not swap the roles of n and p.

  12. Confirm the probability range

    p∈(0,1)p\in(0,1)

    A valid probability lies strictly between 0 and 1.

  13. Confirm n is a positive integer

    n∈Z+n\in\mathbb{Z}^{+}

    The number of trials is a positive integer.

  14. Select the correct model

    ⇒choose B(n,p)\Rightarrow\text{choose }B(n,p)

    Pick the option with the correct n and p.

  15. State the model

    X∼B(60,0.58)X\sim B(60,0.58)

    'Do not vote for the party' has probability 1 - 0.42 = 0.58.

Answer
X∼B(60,0.58)X\sim B(60,0.58)
Question 3
8 markschallenging
A student models the number of rainy days in a 3030-day month as B(30,p)B(30,p) using the long-run proportion of rainy days for pp. Which comment on this model is the most valid criticism?
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Worked solution

  1. Identify the repeated trials

    n repeated trialsn\text{ repeated trials}

    There is a fixed number of repeated trials.

  2. Check each trial has two outcomes

    {success, failure}\{\text{success},\ \text{failure}\}

    Every trial results in exactly one of two outcomes.

  3. Check the probability of success is constant

    p=constantp=\text{constant}

    The probability of success is the same on every trial.

  4. Check the trials are independent

    trials independent\text{trials independent}

    The outcome of one trial does not affect the others.

  5. Check the number of trials is fixed in advance

    n fixedn\text{ fixed}

    n is decided before the experiment begins.

  6. Compare with the binomial requirements

    B(n,p) requirementsB(n,p)\text{ requirements}

    Compare the described experiment with the four binomial conditions.

  7. Define the variable of interest

    X=number of successesX=\text{number of successes}

    Let X be the number of successes.

  8. Note what could break the model

    dependence or changing p\text{dependence or changing }p

    Sampling without replacement or a changing p would break the model.

  9. Assess whether p stays the same

    p1=p2=⋯=pnp_1=p_2=\dots=p_n

    If p changes between trials the model fails.

  10. Assess independence carefully

    P(A∩B)=P(A) P(B)P(A\cap B)=P(A)\,P(B)

    Independent events multiply their probabilities.

  11. Consider sample size versus population

    n≪Nn\ll N

    For sampling without replacement, n must be small relative to N.

  12. Check the two outcomes partition each trial

    success+failure=1\text{success}+\text{failure}=1

    The two outcomes must cover every possibility.

  13. Summarise which conditions hold

    four conditions checked\text{four conditions checked}

    Summarise which conditions are satisfied.

  14. Select the best justification

    ⇒choose the statement\Rightarrow\text{choose the statement}

    Choose the option matching the analysis.

  15. State the conclusion

    clustering⇒independence doubtful\text{clustering}\Rightarrow\text{independence doubtful}

    Weather on consecutive days is correlated, undermining independence.

Answer
Rainy days tend to cluster, so days are not independent and the constant-pp/independence conditions are doubtful.
Question 4
8 markschallenging
For X∼B(28,0.5)X\sim B(28,0.5), find the largest value of rr such that P(X≥r)≥0.85P(X\ge r)\ge 0.85.
Show worked solution

Worked solution

  1. Recognise the binomial model

    fixed n, 2 outcomes, constant p, independent\text{fixed }n,\ 2\text{ outcomes},\ \text{constant }p,\ \text{independent}

    The situation is binomial.

  2. Define the random variable

    X=number of successesX=\text{number of successes}

    Let X count the successes.

  3. State the number of trials

    n=28n=28

    There are n trials.

  4. State the probability of success

    p=0.5p=0.5

    p is the success probability.

  5. Write the distribution

    X∼B(28, 0.5)X\sim B(28,\ 0.5)

    X follows a binomial distribution.

  6. State the condition to satisfy

    largest r: P(X≥r)≥0.85\text{largest }r:\ P(X\ge r)\ge 0.85

    Translate the requirement into an inequality on a cumulative probability.

  7. Choose a search method

    decrease r while P(X≥r)≥0.85\text{decrease }r\text{ while }P(X\ge r)\ge 0.85

    Evaluate cumulative probabilities until the boundary is found.

  8. Evaluate the cumulative probability just before the boundary

    P(X≥11)=0.9075P(X\ge 11)=0.9075

    Compute the cumulative probability at the value just before the boundary.

  9. Evaluate the cumulative probability at the boundary

    P(X≥12)=0.8275P(X\ge 12)=0.8275

    Compute the cumulative probability at the candidate value.

  10. Compare with the threshold

    0.9075≥0.85 but 0.8275<0.850.9075\ge 0.85\text{ but }0.8275<0.85

    Check which value first meets the required inequality.

  11. Identify the boundary value

    r=11r=11

    This is the value where the condition is first met.

  12. Confirm the neighbouring value fails

    P(X≥12)=0.8275<0.85P(X\ge 12)=0.8275<0.85

    Check that the adjacent value does not satisfy the condition.

  13. State the required value

    ⇒r=11\Rightarrow r=11

    State the value of r.

  14. Verify the inequality

    P(X≥11)=0.9075≥0.85P(X\ge 11)=0.9075\ge 0.85

    Confirm the chosen value satisfies the condition.

  15. Conclude

    ∴r=11\therefore r=11

    This is the required smallest/largest value.

Answer
r=11r=11
Question 5
8 markschallenging
For X∼B(18,0.35)X\sim B(18,0.35), find the largest value of rr such that P(X≥r)≥0.8P(X\ge r)\ge 0.8.
Show worked solution

Worked solution

  1. Recognise the binomial model

    fixed n, 2 outcomes, constant p, independent\text{fixed }n,\ 2\text{ outcomes},\ \text{constant }p,\ \text{independent}

    The situation is binomial.

  2. Define the random variable

    X=number of successesX=\text{number of successes}

    Let X count the successes.

  3. State the number of trials

    n=18n=18

    There are n trials.

  4. State the probability of success

    p=0.35p=0.35

    p is the success probability.

  5. Write the distribution

    X∼B(18, 0.35)X\sim B(18,\ 0.35)

    X follows a binomial distribution.

  6. State the condition to satisfy

    largest r: P(X≥r)≥0.8\text{largest }r:\ P(X\ge r)\ge 0.8

    Translate the requirement into an inequality on a cumulative probability.

  7. Choose a search method

    decrease r while P(X≥r)≥0.8\text{decrease }r\text{ while }P(X\ge r)\ge 0.8

    Evaluate cumulative probabilities until the boundary is found.

  8. Evaluate the cumulative probability just before the boundary

    P(X≥5)=0.8114P(X\ge 5)=0.8114

    Compute the cumulative probability at the value just before the boundary.

  9. Evaluate the cumulative probability at the boundary

    P(X≥6)=0.645P(X\ge 6)=0.645

    Compute the cumulative probability at the candidate value.

  10. Compare with the threshold

    0.8114≥0.8 but 0.645<0.80.8114\ge 0.8\text{ but }0.645<0.8

    Check which value first meets the required inequality.

  11. Identify the boundary value

    r=5r=5

    This is the value where the condition is first met.

  12. Confirm the neighbouring value fails

    P(X≥6)=0.645<0.8P(X\ge 6)=0.645<0.8

    Check that the adjacent value does not satisfy the condition.

  13. State the required value

    ⇒r=5\Rightarrow r=5

    State the value of r.

  14. Verify the inequality

    P(X≥5)=0.8114≥0.8P(X\ge 5)=0.8114\ge 0.8

    Confirm the chosen value satisfies the condition.

  15. Conclude

    ∴r=5\therefore r=5

    This is the required smallest/largest value.

Answer
r=5r=5

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