During the winter months in a certain village in Scotland, the probability of a day having severe fog is \(0.6\). The probability that in a given week there will be exactly two days with severe fog is:

Question

Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% of learners were self-taught using internet resources and upskilled themselves.

A student may spend 1 hour to 6 hours in a day in upskilling self. The probability distribution of the number of hours spent by a student is given below:

\[ P(X = x) = \begin{cases} kx^2, & \text{for } x = 1, 2, 3, \\ 2kx, & \text{for } x = 4, 5, 6, \\ 0, & \text{otherwise.} \end{cases} \]

where \( x \) denotes the number of hours. Based on the above information, answer the following questions:
(i) Express the probability distribution given above in the form of a probability distribution table.
(ii) Find the value of \( k \).
(iii)(a) Find the mean number of hours spent by the student.
(iii)(b) Find \( P(1 < X < 6) \).

Answer 1

Step 1: Identify the distribution.
This is binomial with \(n=7\), \(p=3/5\), \(q=2/5\). We need \(P(X=2)\).

Step 2: Apply the formula.
\[P(X=2) = \binom{7}{2}\left(\frac{3}{5}\right)^2\left(\frac{2}{5}\right)^5 = 21 \cdot \frac{9}{25} \cdot \frac{32}{3125} = 21 \cdot \frac{288}{78125} = \frac{6048}{5^7}.\]
\[\boxed{\frac{6048}{5^7}}\]

During the winter months in a certain village in Scotland, the probability of a day having severe fog is \(0.6\). The probability that in a given week there will be exactly two days with severe fog is:

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The Correct Option is A

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