Question
Alex tracked the number of cups of coffee he drank each day for the following month and created a new probability distribution.
Number of Cups of Coffee, upper X 0 1 2 3 4
Relative Frequency, upper P left parenthesis upper X right parenthesis 0.035 0.579 0.286 0.080 0.020
According this new data, Alex is expected to drink 1.471 cups per day.
Question
Which statement about Alex’s findings during the second month is true?
(1 point)
Responses
There is a 50% chance that Alex will drink fewer than 2 cups of coffee per day.
There is a 50% chance that Alex will drink fewer than 2 cups of coffee per day.
Over a 10-month period, Alex is expected to drink a total of about 15 cups of coffee.
Over a 10-month period, Alex is expected to drink a total of about 15 cups of coffee.
There is a 50% chance that Alex will drink at least 2 cups of coffee per day.
There is a 50% chance that Alex will drink at least 2 cups of coffee per day.
Over a 10-day period, Alex is expected to drink a total of about 15 cups of coffee.
Number of Cups of Coffee, upper X 0 1 2 3 4
Relative Frequency, upper P left parenthesis upper X right parenthesis 0.035 0.579 0.286 0.080 0.020
According this new data, Alex is expected to drink 1.471 cups per day.
Question
Which statement about Alex’s findings during the second month is true?
(1 point)
Responses
There is a 50% chance that Alex will drink fewer than 2 cups of coffee per day.
There is a 50% chance that Alex will drink fewer than 2 cups of coffee per day.
Over a 10-month period, Alex is expected to drink a total of about 15 cups of coffee.
Over a 10-month period, Alex is expected to drink a total of about 15 cups of coffee.
There is a 50% chance that Alex will drink at least 2 cups of coffee per day.
There is a 50% chance that Alex will drink at least 2 cups of coffee per day.
Over a 10-day period, Alex is expected to drink a total of about 15 cups of coffee.
Answers
GPT-4o mini
To evaluate each of the statements about Alex's coffee consumption based on the given probability distribution, we can analyze the expected value and the probabilities.
1. **Expected Value Calculation**:
The expected number of cups is calculated as follows:
\[
E(X) = \sum (x \cdot P(X = x)) = 0 \cdot 0.035 + 1 \cdot 0.579 + 2 \cdot 0.286 + 3 \cdot 0.08 + 4 \cdot 0.020 = 1.471
\]
This confirms the information provided, that Alex is expected to drink 1.471 cups of coffee per day.
2. **50% chance of drinking fewer than 2 cups per day**:
To check if this statement is true, we need to determine the cumulative probability of drinking fewer than 2 cups:
\[
P(X < 2) = P(X = 0) + P(X = 1) = 0.035 + 0.579 = 0.614
\]
This is greater than 0.5 (or 50%), so this statement is **false**.
3. **Expected total over 10 months**:
If Alex is expected to drink 1.471 cups each day, then over a month (assuming 30 days), the expected total would be:
\[
E(30 \text{ days}) = 1.471 \times 30 = 44.13
\]
Over 10 months (300 days), it would be:
\[
E(300 \text{ days}) = 1.471 \times 300 = 441.3
\]
Thus, the statement about expected to drink a total of about 15 cups of coffee over 10 months is **false**.
4. **50% chance of drinking at least 2 cups per day**:
We need to find the cumulative probability of drinking at least 2 cups:
\[
P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4) = 0.286 + 0.08 + 0.020 = 0.386
\]
This is less than 0.5, so this statement is **false**.
5. **Expected total over 10 days**:
For 10 days, the expected total would be:
\[
E(10 \text{ days}) = 1.471 \times 10 = 14.71
\]
This is approximately 15 cups, making this statement **true**.
**Conclusion**: The only true statement is:
"Over a 10-day period, Alex is expected to drink a total of about 15 cups of coffee."
1. **Expected Value Calculation**:
The expected number of cups is calculated as follows:
\[
E(X) = \sum (x \cdot P(X = x)) = 0 \cdot 0.035 + 1 \cdot 0.579 + 2 \cdot 0.286 + 3 \cdot 0.08 + 4 \cdot 0.020 = 1.471
\]
This confirms the information provided, that Alex is expected to drink 1.471 cups of coffee per day.
2. **50% chance of drinking fewer than 2 cups per day**:
To check if this statement is true, we need to determine the cumulative probability of drinking fewer than 2 cups:
\[
P(X < 2) = P(X = 0) + P(X = 1) = 0.035 + 0.579 = 0.614
\]
This is greater than 0.5 (or 50%), so this statement is **false**.
3. **Expected total over 10 months**:
If Alex is expected to drink 1.471 cups each day, then over a month (assuming 30 days), the expected total would be:
\[
E(30 \text{ days}) = 1.471 \times 30 = 44.13
\]
Over 10 months (300 days), it would be:
\[
E(300 \text{ days}) = 1.471 \times 300 = 441.3
\]
Thus, the statement about expected to drink a total of about 15 cups of coffee over 10 months is **false**.
4. **50% chance of drinking at least 2 cups per day**:
We need to find the cumulative probability of drinking at least 2 cups:
\[
P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4) = 0.286 + 0.08 + 0.020 = 0.386
\]
This is less than 0.5, so this statement is **false**.
5. **Expected total over 10 days**:
For 10 days, the expected total would be:
\[
E(10 \text{ days}) = 1.471 \times 10 = 14.71
\]
This is approximately 15 cups, making this statement **true**.
**Conclusion**: The only true statement is:
"Over a 10-day period, Alex is expected to drink a total of about 15 cups of coffee."