Let 'X' denote the number of hours you study during a randomly selected school day. The probability that 'X' can take the values of x has the following form, where k is some unknown constant.
`P(X=x)={{:(0.1",","if",x=0),(kx",","if",x=1" or "2),(k(5-x)",","if",x=3" or "4),(0",",,"otherwise.":}`
(a) Find the value of 'k'.
(b) What is the probability that you study at least two hours ? Exactly two hours ? At most two hours ?

Answer

To solve the problem, we will break it down into two parts as specified in the question.

### Part (a): Find the value of 'k'.

1. **Understanding the Probability Distribution**:
We have the following probability distribution:
- \( P(X = 0) = 0.1 \)
- \( P(X = 1) = k \)
- \( P(X = 2) = k \)
- \( P(X = 3) = k(5 - 3) = 2k \)
- \( P(X = 4) = k(5 - 4) = k \)
- \( P(X = x) = 0 \) for other values of \( x \).

2. **Setting up the Equation**:
The sum of all probabilities must equal 1:
\[ P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 1 \]
Substituting the values we have:
\[ 0.1 + k + k + 2k + k = 1 \]
This simplifies to:
\[ 0.1 + 5k = 1 \]

3. **Solving for k**:
Rearranging the equation gives:
\[ 5k = 1 - 0.1 \]
\[ 5k = 0.9 \]
\[ k = \frac{0.9}{5} = 0.18 \]

### Part (b): Finding the probabilities for different scenarios.

1. **Probability of Studying at Least 2 Hours**:
This means we need to find \( P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4) \).
- \( P(X = 2) = k = 0.18 \)
- \( P(X = 3) = 2k = 2 \times 0.18 = 0.36 \)
- \( P(X = 4) = k = 0.18 \)

Therefore,
\[ P(X \geq 2) = 0.18 + 0.36 + 0.18 = 0.72 \]

2. **Probability of Studying Exactly 2 Hours**:
This is simply:
\[ P(X = 2) = k = 0.18 \]

3. **Probability of Studying at Most 2 Hours**:
This means we need to find \( P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \).
- \( P(X = 0) = 0.1 \)
- \( P(X = 1) = k = 0.18 \)
- \( P(X = 2) = k = 0.18 \)

Therefore,
\[ P(X \leq 2) = 0.1 + 0.18 + 0.18 = 0.46 \]

### Summary of Results:
- (a) The value of \( k \) is \( 0.18 \).
- (b)
- Probability of studying at least 2 hours: \( 0.72 \)
- Probability of studying exactly 2 hours: \( 0.18 \)
- Probability of studying at most 2 hours: \( 0.46 \)