Chapter 6, Problem T6.10SPT

To determine

To choose: The correct option for the probability that the person guesses the first 4 shapes incorrectly but gets fifth one right.

Answer to Problem T6.10SPT

Option c is the correct choice.

Explanation of Solution

Given information:

Number of shapes $=\text{ 5}$

There is an equal chance of choosing each shape.

Number of incorrect guesses $=\text{ 4}$

Concept used:

If the trials are repeated until a success occurs and the probability of success is same in each trial, then it is the case of geometric distribution.

In this case, the shape was correctly guessed in the fifth place and the probability of guessing each shape correctly was same in each trial and was independent. So, this follows geometric distribution.

Calculation:

Let X be the number of trials for the first success.

Probability of guessing a shape correctly $=\frac{1}{\text{Number of shapes}}=\frac{1}{5}$ .

So, $\text{X }~\text{ Geometric with p = }\left(\frac{1}{5}\right)$

  $P(X=k)={{\displaystyle (1-p)}}^{k-1}\times p$

Probability of guessing 4 shapes incorrectly and the fifth one correctly

  $\begin{array}{l}P(X=5)={{\displaystyle \left(1-\frac{1}{5}\right)}}^{5-1}\times \frac{1}{5}\\ P(X=5)={{\displaystyle \left(\frac{4}{5}\right)}}^4\times \frac{1}{5}\end{array}$