John and Anna each shoot three times, with independent hit probabilities 1/3 and 2/3. Find the probability of four total hits.

01 / Standard#Your turn

Official paper · jm01-2025 · I.12 · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

List the possible hit-count pairs.

Hint 2

Use (1,3), (2,2), and (3,1).

Worked solution

Calculate the required binomial probabilities for each person.
$P_J(1,2,3)=\left(\frac49,\frac29,\frac1{27}\right),\quad P_A(1,2,3)=\left(\frac29,\frac49,\frac8{27}\right)$
Multiply within each independent pair, then add.
$P=\frac49\frac8{27}+\frac29\frac49+\frac1{27}\frac29=\frac{58}{243}$

D: 58/243.

Checks and common pitfalls: The six shots do not share one success probability.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Calculate the required binomial probabilities for each person.
$P_J(1,2,3)=\left(\frac49,\frac29,\frac1{27}\right),\quad P_A(1,2,3)=\left(\frac29,\frac49,\frac8{27}\right)$
Multiply within each independent pair, then add.
$P=\frac49\frac8{27}+\frac29\frac49+\frac1{27}\frac29=\frac{58}{243}$

Think first. Reveal a hint when the class is ready.

2025 JM01