Question

Ellie and Erick assert that they could communicate telepathically. To test this assertion, we place these two friends in separate rooms and show Ellie a series of cards. Each card is equally likely to depict either a circle or a triangle or a rectangle. For each card presented to Ellie. Erick writes down circle'. or 'triangle or 'rectangle', depending on what he believes Ellie to be looking at. If ten cards are shown, what is the probability that Erick correctly matches at least one?

Answer 1

A match will occur if Erick writes down the right card, i.e, Erick writes cicle an Ellie gets a circle, Erick writes triangle and Ellie gets a triangle or Erick writes a rectangle and Ellie gets a rectangle.

Thus, Pr[A match] = Pr[Erick writes circle and Ellie gets a circle $\cup$ Erick writes triangle and Ellie gets a triangle $\cup$ Erick writes rectangle and Ellie gets a rectangle]

= Pr[Erick writes circle and Ellie gets a circle] + Pr[Erick writes triangle and Ellie gets a triangle] + Pr[Erick writes rectangle and Ellie gets a rectangle]

= 1/9 + 1/9 + 1/9

= 3/9

= 1/3

We are to find the probability of atleast one match in 10 cards.

Let X be a random variable such that X = The number of matches in 10 cards.

Thus, X ~ bin(10, 1/3)

Thus, P[X = x] = , x = 0,1,...,10.

10-r (0) ) (1– ) o-

So, we want Pr[ X $\geq$ 1]

Pr[ X $\geq$ 1]

= 1 - Pr[X<1]

= 1 - Pr[X = 0]

= 1 - $\binom{10}{0}(\frac{1}{3})^{0}(1 - \frac{1}{3})^{10 - 0}$

= 1 - $( \frac{2}{3})^{10}$

= 0.9826 (Ans.)