Answer to Question #176404 in Statistics and Probability for Ananya

Question

Suppose twin brothers, Lim and Jim are taking part in an experiment called ‘Twin’s

Telepathy Test’. In this test, 5 candles of different colors are shown to both the brothers

in the beginning. Then, the twins are put into two different rooms and the candles are

given to Jim. He picks a candle himself, presses a buzzer to let Lim know that one

candle’s been picked without telling him the color of the candle. Then Jim tries to

connect to Lim through his mind and transmit the name of the color to Lim telepathically.

Now, Lim has to guess the color of the candle. This goes on for 5 times for all five

candles.

a. What is the probability that Lim makes exactly 4 right guesses?

b. The same experiment is repeated, but this time after each wrong guess Lim makes,

the correct answer is shown to him. Now what is the probability that Lim makes

exactly 4 right guesses?

Answer 1

a. Probability that Lim makes a right guess is $\frac{1}{5}$

Probability that Lim makes exactly 4 right guesses can be calculated using Binomial trials

n=5 and $p= \frac{1}{5}$

Let, X be the number of right guesses out of 5

$\small P(X=4)=\;^5C_4\times(\frac{1}{5})^4\times(\frac{4}{5})^1=\frac{4}{625}$

b. Now, Lim is known the right answer. So, after each time is experiment is conducted, the number of candles he has to guess gets decreased by 1.

For first experiment, probability of guessing right $= \frac{1}{5}$

For 2nd experiment $= \frac{1}{4}$

For 3rd experiment $= \frac{1}{3}$

For 4th experiment $= \frac{1}{2}$

For 5th experiment = 1

Now, it is certain that he will get the 5th guess right.

In addition, at least one of the 1 to 4th guess should be wrong. Which will have probability

Probability that only first answer is wrong

$\small =\frac{4}{5}\times\frac{1}{4}\times\frac{1}{3}\times\frac{1}{2}=\frac{4}{120}$

Probability that only 2nd guess is wrong

$\small =\frac{1}{5}\times\frac{3}{4}\times\frac{1}{3}\times\frac{1}{2}=\frac{3}{120}$

Probability that only 3rd guess is wrong

$\small =\frac{1}{5}\times\frac{1}{4}\times\frac{2}{3}\times\frac{1}{2}=\frac{2}{120}$

Probability that only 2nd guess is wrong

$\small =\frac{1}{5}\times\frac{1}{4}\times\frac{1}{3}\times\frac{1}{2}=\frac{1}{120}$

Required probability

$\small =\frac{4}{120}+\frac{3}{120}+\frac{2}{120}+\frac{1}{120}=\frac{1}{12}$