Question

A box contains 7 letters, shown as TRACKED. what is the probability of the outcome TRACE in that order if 5 letters are drawn one by one (a) with replacement? and (b) without replacement?

Answer 1

There are 7 letters in a box, shown as TRACKED. We have to pick up 5 letters in the order such that the outcome is TRACE.

a)

Let each letter is picked with replacement.

So, the probability of picking up letter T first is,

$P(T)=\frac{1}{7}$

We replaced the picked up letter T. So, the number of letters 7 in the box remains the same. Now, the probability of picking up letter R is,

$P(R)=\frac{1}{7}$

Similarly, the probability of picking up letters A,C and E respectively with replacement is,

$\begin{gathered} P\mleft(A\mright)=\frac{1}{7} \\ P(C)=\frac{1}{7} \\ P(E)=\frac{1}{7} \end{gathered}$

Hence, the probability of the outcome TRACE in that order if 5 letters are drawn one by one with replacement is,

$\begin{gathered} P(\text{with replacement)=P(T)}\cdot\text{P(R)}\cdot P(A)\cdot P(C)\cdot P(E) \\ =\frac{1}{7}.\frac{1}{7}.\frac{1}{7}.\frac{1}{7}.\frac{1}{7} \\ =\frac{1}{16809} \end{gathered}$

So, the probability of the of the outcome TRACE in that order if 5 letters are drawn one by one with replacement is 1/16809.

b)

Let each letter in TRACE is picked in order without replacement.

So, the probability of picking up letter T first is,

$P(T)=\frac{1}{7}$

Now, there are only 6 letters remaining in the box. So, the probability of picking up letters R without replacement is,

$P(R)=\frac{1}{6}$

Similarly, the probability of picking up letters A,C and E respectively without replacement is,

$\begin{gathered} P(A)=\frac{1}{5} \\ P(C)=\frac{1}{4} \\ P(E)=\frac{1}{3} \end{gathered}$

Now, the probability of the outcome TRACE in that order if 5 letters are drawn one by one without replacement is,

$\begin{gathered} P(withoutreplacement)=P(T)\cdot P(R)\cdot P(A)\cdot P(C)\cdot P(E) \\ =\frac{1}{7}\cdot\frac{1}{6}\cdot\frac{1}{5}\cdot\frac{1}{4}\cdot\frac{1}{3} \\ =\frac{1}{2520} \end{gathered}$

Therefore, the probability of the outcome TRACE in that order if 5 letters are drawn one by one without replacement is 1/2520.