8. Gallant has a subscription to "free coffee" at Soakin' Bagels, but he still often tips the employees. (Goofus pays full price for coffee and does not tip.) If he doesn't tip it's usually because he doesn't have any $1 bills on hand. The following table gives the pmf of X, Gallant's tip each time he goes to Soakin' Bagels. Table 10.5.2. $0 $1$2 $3 P[Xx] 0.1 0.5 0.35 0.05 I a. Verify that the above pmf is indeed a pmf. b. What is the probability that Gallant tips? c. What is the average value of Gallant's tip each morning?
8. Gallant has a subscription to "free coffee" at Soakin' Bagels, but he still often tips the employees. (Goofus pays full price for coffee and does not tip.) If he doesn't tip it's usually because he doesn't have any $1 bills on hand. The following table gives the pmf of X, Gallant's tip each time he goes to Soakin' Bagels. Table 10.5.2. $0 $1$2 $3 P[Xx] 0.1 0.5 0.35 0.05 I a. Verify that the above pmf is indeed a pmf. b. What is the probability that Gallant tips? c. What is the average value of Gallant's tip each morning?
A First Course in Probability (10th Edition)
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![**Problem 8 Explanation:**
Gallant has a "free coffee" subscription at Soakin' Bagels, but he usually tips the employees. If he doesn't tip, it's generally because he doesn't have any $1 bills available. The table given describes the probability mass function (pmf) for \( X \), which represents Gallant's tip every time he visits Soakin' Bagels.
**Table 10.5.2: Probability Mass Function of Gallant's Tip**
The pmf table is structured as follows:
- \( x \) represents the amount Gallant tips, which can be $0, $1, $2, or $3.
- \( P[X = x] \) shows the probabilities corresponding to each tip amount:
- Probability of tipping $0: 0.1
- Probability of tipping $1: 0.5
- Probability of tipping $2: 0.35
- Probability of tipping $3: 0.05
**Questions:**
a. **Verify that the given pmf is indeed a pmf.**
To verify, ensure that the sum of probabilities equals 1:
\[ 0.1 + 0.5 + 0.35 + 0.05 = 1 \]
b. **What is the probability that Gallant tips?**
Gallant tips if \( X \) is $1, $2, or $3. Calculate this probability by summing the relevant probabilities:
\[ P[X = 1] + P[X = 2] + P[X = 3] = 0.5 + 0.35 + 0.05 = 0.9 \]
c. **What is the average value of Gallant's tip each morning?**
The average tip, or expected value \( E(X) \), is calculated using the formula:
\[ E(X) = \sum (x \cdot P[X = x]) \]
\[ E(X) = 0 \cdot 0.1 + 1 \cdot 0.5 + 2 \cdot 0.35 + 3 \cdot 0.05 \]
\[ E(X) = 0 + 0.5 + 0.7 + 0.15 = 1.35 \]
The average tip Gallant leaves each morning is $1.35.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7cef3050-b0aa-40af-9112-57cab04a18ba%2F10dc0e4b-f011-4efe-80c8-eb756a02634a%2Fy15j8j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 8 Explanation:**
Gallant has a "free coffee" subscription at Soakin' Bagels, but he usually tips the employees. If he doesn't tip, it's generally because he doesn't have any $1 bills available. The table given describes the probability mass function (pmf) for \( X \), which represents Gallant's tip every time he visits Soakin' Bagels.
**Table 10.5.2: Probability Mass Function of Gallant's Tip**
The pmf table is structured as follows:
- \( x \) represents the amount Gallant tips, which can be $0, $1, $2, or $3.
- \( P[X = x] \) shows the probabilities corresponding to each tip amount:
- Probability of tipping $0: 0.1
- Probability of tipping $1: 0.5
- Probability of tipping $2: 0.35
- Probability of tipping $3: 0.05
**Questions:**
a. **Verify that the given pmf is indeed a pmf.**
To verify, ensure that the sum of probabilities equals 1:
\[ 0.1 + 0.5 + 0.35 + 0.05 = 1 \]
b. **What is the probability that Gallant tips?**
Gallant tips if \( X \) is $1, $2, or $3. Calculate this probability by summing the relevant probabilities:
\[ P[X = 1] + P[X = 2] + P[X = 3] = 0.5 + 0.35 + 0.05 = 0.9 \]
c. **What is the average value of Gallant's tip each morning?**
The average tip, or expected value \( E(X) \), is calculated using the formula:
\[ E(X) = \sum (x \cdot P[X = x]) \]
\[ E(X) = 0 \cdot 0.1 + 1 \cdot 0.5 + 2 \cdot 0.35 + 3 \cdot 0.05 \]
\[ E(X) = 0 + 0.5 + 0.7 + 0.15 = 1.35 \]
The average tip Gallant leaves each morning is $1.35.
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