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Question 1
The discrete random variable $X$ has the following probability distribution:

$x$ 1 2 3
$P (X = x)$ $3 k$ $k$ $0.3$

Find the value of $k$ .
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $k = 0.175$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: State the condition for a valid probability distribution
All probabilities must sum to 1: $\sum P (X = x) = 1$ , so $3 k + k + 0.3 = 1$ .
Step 2: Simplify and solve for $k$
Combining like terms: $4 k + 0.3 = 1$ , giving $4 k = 0.7$ , so $k = 0.175$ .
Step 3: Verify the solution
Check: $3 (0.175) + 0.175 + 0.3 = 0.525 + 0.175 + 0.3 = 1$ ✓. The answer is $k = 0.175$ .
Method #2Approach 2
Step 1: Identify what is needed
We need $3 k + k + 0.3 = 1$ , i.e. $4 k = 0.7$ , so $k = 0.175$ .
Step 2: Eliminate $k = 0.7$
If $k = 0.7$ , then $3 (0.7) + 0.7 + 0.3 = 2.1 + 0.7 + 0.3 = 3.1 \neq = 1$ . This is invalid.
Step 3: Eliminate $k = 0.25$
If $k = 0.25$ , then $3 (0.25) + 0.25 + 0.3 = 0.75 + 0.25 + 0.3 = 1.3 \neq = 1$ . This is invalid.
Step 4: Eliminate $k = 0.1$
If $k = 0.1$ , then $3 (0.1) + 0.1 + 0.3 = 0.3 + 0.1 + 0.3 = 0.7 \neq = 1$ . This is invalid.
Step 5: Select the correct answer
Only $k = 0.175$ satisfies $4 k = 0.7$ , giving probabilities that sum to 1.
Question 2
A discrete random variable $W$ has the probability distribution shown below:

$w$ 0 1 2 3
$P (W = w)$ $0.15$ $0.40$ $m$ $0.05$

What is the value of $m$ ?
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $m = 0.40$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Use the sum-to-one rule
For any valid probability distribution, all probabilities must sum to 1: $0.15 + 0.40 + m + 0.05 = 1$ .
Step 2: Solve for $m$
Adding the known values: $0.60 + m = 1$ , so $m = 0.40$ .
Step 3: Confirm validity
Since $0 \leq 0.40 \leq 1$ , this is a valid probability. The answer is $m = 0.40$ .
Method #2Approach 2
Step 1: Identify the constraint
The known probabilities sum to $0.15 + 0.40 + 0.05 = 0.60$ , so $m = 1 - 0.60 = 0.40$ .
Step 2: Eliminate $m = 0.35$
If $m = 0.35$ , total $= 0.60 + 0.35 = 0.95 \neq = 1$ . Not valid.
Step 3: Eliminate $m = 0.45$
If $m = 0.45$ , total $= 0.60 + 0.45 = 1.05 \neq = 1$ . Exceeds 1, so invalid.
Step 4: Eliminate $m = 0.30$
If $m = 0.30$ , total $= 0.60 + 0.30 = 0.90 \neq = 1$ . Not valid.
Step 5: Select the correct answer
Only $m = 0.40$ makes the probabilities sum to exactly 1.

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14 more questions in this topic

x

P (X = x)

3 k

k

0.3

w

P (W = w)

0.15

0.40

m

0.05

SL 4.7—Discrete random variables

Step-by-step walkthrough

Step 1: State the condition for a valid probability distribution

Step 2: Simplify and solve for $k$

Step 3: Verify the solution

Step 1: Identify what is needed

Step 2: Eliminate $k = 0.7$

Step 3: Eliminate $k = 0.25$

Step 4: Eliminate $k = 0.1$

Step 5: Select the correct answer

Step-by-step walkthrough

Step 1: Use the sum-to-one rule

Step 2: Solve for $m$

Step 3: Confirm validity

Step 1: Identify the constraint

Step 2: Eliminate $m = 0.35$

Step 3: Eliminate $m = 0.45$

Step 4: Eliminate $m = 0.30$

Step 5: Select the correct answer

14 more questions in this topic

SL 4.7—Discrete random variables

Step-by-step walkthrough

Step 1: State the condition for a valid probability distribution

Step 2: Simplify and solve for $k$

Step 3: Verify the solution

Step 1: Identify what is needed

Step 2: Eliminate $k = 0.7$

Step 3: Eliminate $k = 0.25$

Step 4: Eliminate $k = 0.1$

Step 5: Select the correct answer

Step-by-step walkthrough

Step 1: Use the sum-to-one rule

Step 2: Solve for $m$

Step 3: Confirm validity

Step 1: Identify the constraint

Step 2: Eliminate $m = 0.35$

Step 3: Eliminate $m = 0.45$

Step 4: Eliminate $m = 0.30$

Step 5: Select the correct answer

14 more questions in this topic