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Problem 207

Combinatorics Difficulty 1.6 Multiple choice CEMC Gauss (Grade 8) · Canada · 2022

The results of asking 50 students if they participate in music or
sports are shown in the Venn diagram.

What percentage of the 50 students do not participate in music and do
not participate in sports?

Pick one

Official solution

Solution 1

When a standard fair coin is tossed three times, there are 8 possible
outcomes.

If H represents head and T represents tail, these 8 outcomes are: HHH,
HHT, HTH, THH, HTT, THT, TTH, and TTT.

Of these 8 outcomes, there are exactly 2 whose outcomes are all the same
(HHH and TTT).

Therefore, the probability that the three outcomes are all the same is
$\frac28=\frac14$ .

Solution 2

The three outcomes are all the same exactly when each of the three
tosses is a head or when each of the three tosses is a tail.

When a coin is tossed, the probability that the coin shows a head is
$\frac12$ and the probability that
the coin shows a tail is also $\frac12$ (there are two possible outcomes
and each is equally probable).

Thus, the probability that each of the three tosses is a head is $\frac12\times\frac12\times\frac12=\frac18$ .

Similarly, the probability that each of the three tosses is a tail is
$\frac12\times\frac12\times\frac12=\frac18$ .

Therefore, the probability that the three outcomes are all the same
is $\frac18+\frac18=\frac28=\frac14$ .

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