Love Each Other

Probability Level 3

One example of those events where everyone goes out on a date

Three males A, B, C and three females X, Y, Z are participating in a weekly match-making TV show Love Each Other .

At the end of the show, they each push one of the 3 buttons in front of them simultaneously to choose the person they want to date. Only those couples who mutually choose each other can go out on a date sponsored by the TV program.

Assuming equal probability of choosing each potential partner, what is the probability that everyone goes out on a date this week?

\frac{3!}{3^6}

\frac{2^3}{3^3}

\frac{(3!)^2}{3^6}

\frac12 \times\frac{3!}{3^3}

1 solution

Jimin Khim Staff
Nov 22, 2017

Each of the 6 participants has 3 people to choose from, so the number of all possible events is $3^6$ with equal probabilities. Then, there are $3!=6$ events in which all of them are fully paired up: $(A\rightleftharpoons X, B\rightleftharpoons Y, C\rightleftharpoons Z),\ (A\rightleftharpoons X, B\rightleftharpoons Z, C\rightleftharpoons Y),\ (A\rightleftharpoons Y, B\rightleftharpoons X, C\rightleftharpoons Z),\ (A\rightleftharpoons Y, B\rightleftharpoons Z, C\rightleftharpoons X),\ (A\rightleftharpoons Z, B\rightleftharpoons X, C\rightleftharpoons Y),\ (A\rightleftharpoons Z, B\rightleftharpoons Y, C\rightleftharpoons X).$

Therefore, the correct answer is $\frac{3!}{3^6}=\frac{2}{243}\approx 0.82\%.\ _\square$

Love Each Other

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