Question

Alex tracked how many cups of coffee he drank each day for a month and then compiled the results into a probability distribution, as shown below.

Number of Cups of Coffee, upper X 0 1 2 3 4
Relative Frequency, upper P left parenthesis upper X right parenthesis 0.024 0.423 0.291 0.172 0.090
Question
Determine the median of Alex’s probability distribution.

(1 point)
Responses

1 cup
1 cup

2 cups
2 cups

2.5 cups
2.5 cups

3 cups

Answer 1

To find the median of the probability distribution, we need to determine the value of \( X \) such that the cumulative probability is at least 0.5.

Here are the details provided in the distribution:

- \( P(X = 0) = 0.024 \)
- \( P(X = 1) = 0.423 \)
- \( P(X = 2) = 0.291 \)
- \( P(X = 3) = 0.172 \)
- \( P(X = 4) = 0.090 \)

Next, we will calculate the cumulative probabilities:

- Cumulative probability for \( X = 0 \):
\[
P(X \leq 0) = P(X = 0) = 0.024
\]

- Cumulative probability for \( X = 1 \):
\[
P(X \leq 1) = P(X = 0) + P(X = 1) = 0.024 + 0.423 = 0.447
\]

- Cumulative probability for \( X = 2 \):
\[
P(X \leq 2) = P(X \leq 1) + P(X = 2) = 0.447 + 0.291 = 0.738
\]

Now let's find where the cumulative probability first reaches or exceeds 0.5:

1. Cumulative probability for \( X = 0 \) is \( 0.024 \) (not enough).
2. Cumulative probability for \( X = 1 \) is \( 0.447 \) (still not enough).
3. Cumulative probability for \( X = 2 \) is \( 0.738 \) (this exceeds 0.5).

Since \( P(X \leq 1) = 0.447 \) is less than 0.5 and \( P(X \leq 2) = 0.738 \) is greater than 0.5, the median value is \( 2 \) cups.

**Answer**: 2 cups