Question

use the two - way table below to answer the questions.
you survey friends about the type of party they enjoy most.

	gender	gender	gender
party type	bowling	6	2	8
party type	skating	3	11	14
party type	dancing	1	3	4
party type	total	10	16	26

how many people enjoy a bowling party or a dancing party?
○ 22
○ 26
○ 18
○ 12
14 numeric 3 points
a card is drawn at random from an ordinary deck of cards. if the card selected is a face card, then amy wins $12. if the card selected is an ace, then amy wins $18. otherwise, she loses $10. what is her expected profit or loss? round to the nearest penny
answer

Explanation:

First Question (Two - way Table)

Step1: Identify totals for bowling and dancing

From the table, total for bowling party is 8, total for dancing party is 4.

Step2: Add the two totals

To find the number of people who enjoy bowling or dancing party, we add the totals: \(8 + 4=12\)? Wait, no, wait. Wait, the "or" here, are the events mutually exclusive? Since a person can't be in both bowling and dancing party (assuming one favorite), so we add the totals. Wait, bowling total is 8, dancing total is 4. So \(8 + 4 = 12\)? But wait, let's check the table again. Bowling: 8, Dancing: 4. So 8+4 = 12. But wait, the options include 12. Wait, but let's re - check. The table: Bowling total is 8, Dancing total is 4. So 8 + 4=12.

Step1: Determine probabilities

A standard deck has 52 cards.

• Face cards: Jack, Queen, King. There are 3 face cards per suit, 4 suits, so \(3\times4 = 12\) face cards. Probability of face card, \(P(F)=\frac{12}{52}\).
• Aces: 4 aces. Probability of ace, \(P(A)=\frac{4}{52}\).
• Remaining cards: \(52-(12 + 4)=36\) cards. Probability of other cards, \(P(O)=\frac{36}{52}\).

Step2: Calculate expected value

The formula for expected value \(E(X)=\sum xP(x)\), where \(x\) is the payout and \(P(x)\) is the probability.

• For face card: \(x = 12\), \(P(x)=\frac{12}{52}\), so contribution is \(12\times\frac{12}{52}\)
• For ace: \(x = 18\), \(P(x)=\frac{4}{52}\), so contribution is \(18\times\frac{4}{52}\)
• For other cards: \(x=- 10\), \(P(x)=\frac{36}{52}\), so contribution is \(- 10\times\frac{36}{52}\)

Step3: Compute each term

• Face card term: \(12\times\frac{12}{52}=\frac{144}{52}\approx2.7692\)
• Ace term: \(18\times\frac{4}{52}=\frac{72}{52}\approx1.3846\)
• Other cards term: \(- 10\times\frac{36}{52}=\frac{-360}{52}\approx - 6.9231\)

Step4: Sum the terms

\(E(X)=\frac{144 + 72-360}{52}=\frac{216 - 360}{52}=\frac{-144}{52}\approx - 2.77\) (rounded to nearest penny)

Answer:

12 (the option with 12)

Second Question (Expected Value)