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Problem 239

Combinatorics Difficulty 1.4 Multiple choice CEMC Cayley · Canada · 2025

Farhan has a blue hat, a white hat, a blue scarf, a white scarf,
and a green scarf. He randomly chooses one hat and one scarf. What is
the probability that the hat and the scarf are the same colour?

Pick one

Official solution

The probability that Farhan chooses a hat and a scarf having the
same colour is equal to the probability that he chooses the white hat
and the white scarf added to the probability that he chooses the blue
hat and the blue scarf.

Since there are $2$ hats, the
probability that he chooses the blue hat is $\frac12$ .

Since there are $3$ scarves, the
probability that he chooses the blue scarf is $\frac13$ .

The probability that Farhan chooses both the blue hat and the blue scarf
is $\frac12\times\frac13=\frac16$ .

Similarly, the probability that Farhan chooses both the white hat and
the white scarf is $\frac16$ .

The probability that Farhan chooses a hat and a scarf having the same
colour is $\frac16+\frac16=\frac26=\frac13$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.