Question

The probability that Vikram is late for school in the morning is \[0.1\]. How many times would you expect Vikram to be on time in 20 mornings?
(a) 2
(b) 9
(c) 18
(d) 1

Answer 1

Hint:
Here, we need to find the number of times that Vikram will be on time in 20 mornings. We will use the probability of an event and its complement to find the probability that Vikram is on time in the morning. Then, using expectations, we will find the expected number of times Vikram is on time in 20 mornings.

Complete step by step solution:
Let \[X\] be the event that Vikram is late for school in the morning.
Therefore, we get
\[P\left( X \right) = 0.1\]
Let \[Y\] be the event that Vikram is on time in the morning.
Now, Vikram can either be on time in the morning, or late.
Therefore, the event that Vikram is on time is complement to the event that Vikram is late for school.
The sum of probabilities of an event and its complement is 1.
Therefore, we get
\[P\left( X \right) + P\left( Y \right) = 1\]
Substituting \[P\left( X \right) = 0.1\] in the equation, we get
\[ \Rightarrow 0.1 + P\left( Y \right) = 1\]
Subtracting \[0.1\] from both sides of the equation, we get
$ \Rightarrow 0.1 + P\left( Y \right) - 0.1 = 1 - 0.1 \\
   \Rightarrow P\left( Y \right) = 0.9 \\ $
We will use expectations to find the expected number of times Vikram is on time in 20 mornings.
Thus, we get
Number of times Vikram is on time in \[y\] mornings \[ = y \times P\left( Y \right)\], where \[y\] is the number of mornings, and \[P\left( Y \right)\] is the probability that Vikram is on time in the morning.
Substituting \[y = 20\] and \[P\left( Y \right) = 0.9\] in the equation, we get
\[ \Rightarrow \]Number of times Vikram is on time in 20 mornings \[ = 20 \times 0.9\]
Multiplying the terms of the expression, we get
\[ \Rightarrow \]Number of times Vikram is on time in 20 mornings \[ = 18\]
Therefore, the expected number of times that Vikram is on time in 20 mornings, is 18.

Thus, the correct option is option (c).

Note:
A common mistake is to multiply 20 by the probability given in the question, that is \[0.1\]. This results in the expected number of times as 2, which is the expected number of times that Vikram will be late in 20 mornings. Thus, the answer will be incorrect.