Question

in new york state, 48% of all teenagers own a skateboard, and 39% of all teenagers own a skateboard and roller blades. we want to find the probability that a teenager owns roller blades given that the teenager owns a skateboard.
a. lets first define events a and b.
event a = teenager own a \boxed{}, event b = teenager own a \boxed{}
b. identify probabilities for each event. p(a) = \boxed{}, p(a \cap b) = \boxed{}
write your answer as a decimal.
c. using the formula below, find the probability that a teenager owns roller blades given that the teenager owns a skateboard?
\boxed{p(b|a) = \frac{p(b \cap a)}{p(a)}} p(b|a) = \boxed{}
round your answer as a decimal rounded to the nearest hundredth.

Explanation:

Part a: Define Events

We need to define events A and B based on the problem. The problem is about finding the probability that a teenager owns roller blades given they own a skateboard. So, Event A should be the event that a teenager owns a skateboard, and Event B should be the event that a teenager owns roller blades.

Part b: Identify Probabilities

Step 1: Find \( P(A) \)

The problem states that 48% of all teenagers own a skateboard. To convert a percentage to a decimal, we divide by 100. So, \( P(A) = \frac{48}{100} = 0.48 \).

Step 2: Find \( P(A \cap B) \)

The problem states that 39% of all teenagers own both a skateboard and roller blades. Converting this percentage to a decimal, we get \( P(A \cap B) = \frac{39}{100} = 0.39 \).

Part c: Calculate Conditional Probability

Step 1: Recall the Conditional Probability Formula

The formula for conditional probability is \( P(B|A) = \frac{P(A \cap B)}{P(A)} \).

Step 2: Substitute the Values

We know \( P(A \cap B) = 0.39 \) and \( P(A) = 0.48 \). Substituting these values into the formula, we get:

$$ P(B|A) = \frac{0.39}{0.48} $$

Step 3: Perform the Division

Calculating \( \frac{0.39}{0.48} \):

$$ \frac{0.39}{0.48} = 0.8125 $$

Step 4: Round to the Nearest Hundredth

Rounding 0.8125 to the nearest hundredth (two decimal places), we look at the third decimal place, which is 2. Since 2 < 5, we round down. So, 0.8125 rounded to the nearest hundredth is 0.81.

Answer:

s:

Part a:

Event A = teenager own a \(\boldsymbol{\text{skateboard}}\), Event B = teenager own a \(\boldsymbol{\text{roller blades}}\)

Part b:

\( P(A) = \boldsymbol{0.48} \), \( P(A \cap B) = \boldsymbol{0.39} \)

Part c:

\( P(B|A) = \boldsymbol{0.81} \) (rounded to the nearest hundredth)