Question

13 the grand hawaiian hotel has 1000 rooms. use the table to the right to determine the probabilities of each problem below. a: room with a king - sized bed b: room with an ocean view a) p(b) = b) p(not a) = c) p(a and b) = d) p(a ∪ b) = 14. you surveyed people about their dessert preference. a) p(cake ∩ ice cream) \\(\frac{23}{80} \approx 0.23\\) b) are the preferences of cake and ice cream independent?

Explanation:

For Problem 1.3

Step1: Define total rooms

Total rooms = 1000

Step2: Calculate P(B)

Rooms with ocean view = 380 + 50 = 430
$P(B) = \frac{430}{1000}$

Step3: Calculate P(Not A)

Rooms without king bed = 380
$P(\text{Not }A) = \frac{380}{1000}$

Step4: Calculate P(A and B)

Rooms with king bed + ocean view = 380
$P(A \text{ and } B) = \frac{380}{1000}$

Step5: Calculate P(A∪B)

Use formula $P(A \cup B) = P(A) + P(B) - P(A \text{ and } B)$
$P(A) = \frac{620}{1000}$, so:
$P(A \cup B) = \frac{620}{1000} + \frac{430}{1000} - \frac{380}{1000} = \frac{670}{1000}$

Step1: Find total surveyed people

Total = $31 + 23 + 17 + 9 = 80$

Step2: Calculate P(cake ∩ ice cream)

$P(\text{cake} \cap \text{ice cream}) = \frac{23}{80} = 0.2875$

Step3: Check independence

First find $P(\text{cake}) = \frac{23+17}{80} = \frac{40}{80} = 0.5$
$P(\text{ice cream}) = \frac{31+23}{80} = \frac{54}{80} = 0.675$
For independence, $P(\text{cake} \cap \text{ice cream}) = P(\text{cake}) \times P(\text{ice cream})$
$0.5 \times 0.675 = 0.3375
eq 0.2875$

Answer:

a) $P(B) = 0.43$
b) $P(\text{Not }A) = 0.38$
c) $P(A \text{ and } B) = 0.38$
d) $P(A \cup B) = 0.67$

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For Problem 1.4