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)=(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)=(1)/(7)`

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

`\begin{gathered} P\mleft(A\mright)=(1)/(7) \\ P(C)=(1)/(7) \\ P(E)=(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) \\ =(1)/(7).(1)/(7).(1)/(7).(1)/(7).(1)/(7) \\ =(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)=(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)=(1)/(6)`

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

`\begin{gathered} P(A)=(1)/(5) \\ P(C)=(1)/(4) \\ P(E)=(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) \\ =(1)/(7)\cdot(1)/(6)\cdot(1)/(5)\cdot(1)/(4)\cdot(1)/(3) \\ =(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.