Question

A student studies for (X) number of hours during a randomly selected school day. The probability that (X) can take the values, has the following form, where (k) is some constant.
(P(X = x) = 0.2, & if x = 0
kx, & if x = 1 or 2
k(6 - x), & if x = 3 or 4
0, & otherwise )
The probability that the student studies for at most two hours is

• A. (0.1)
• B. (0.5)
• C. (0.3)
• D. [suspicious link removed]

Hint:

Always sum all possible probabilities to 1 to find the unknown constant $k$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem defines a discrete probability distribution.
We need to find the value of constant \(k\) first and then find \(P(X \leq 2)\).
Step 2: Key Formula or Approach:
The sum of all probabilities in a probability distribution must be equal to 1: \(\sum P(X=x) = 1\).
Step 3: Detailed Explanation:
Write down the individual probabilities:
\(P(0) = 0.2\)
\(P(1) = k(1) = k\)
\(P(2) = k(2) = 2k\)
\(P(3) = k(6 - 3) = 3k\)
\(P(4) = k(6 - 4) = 2k\)
Summing them up:
\[ 0.2 + k + 2k + 3k + 2k = 1 \]
\[ 0.2 + 8k = 1 \]
\[ 8k = 0.8 \implies k = 0.1 \]
The probability of studying for "at most two hours" is \(P(X \leq 2)\):
\[ P(X \leq 2) = P(0) + P(1) + P(2) \]
\[ P(X \leq 2) = 0.2 + k + 2k = 0.2 + 3k \]
Substitute \(k = 0.1\):
\[ P(X \leq 2) = 0.2 + 3(0.1) = 0.2 + 0.3 = 0.5 \]
Step 4: Final Answer:
The probability is \(0.5\).