Problem 27 - Algebra 2

Question

Anita scores well enough at a carnival game that she gets to randomly draw two prizes out a prize bag. There are 6 purple T-shirts, 8 yellow T-shirts, and 5 T-shirts with a picture of a celebrity on them in the bag. Find each probability. \(P(\text { choosing a celebrity, then a yellow) }\)

Step-by-Step Solution

Answer

The probability is \(\frac{20}{171}\).

Step 1: Determine Total Number of T-Shirts

First, count the total number of T-shirts in the prize bag. There are 6 purple T-shirts, 8 yellow T-shirts, and 5 celebrity T-shirts. Add these together to find the total: \(6 + 8 + 5 = 19\) T-shirts.

Step 2: Calculate Probability of Drawing a Celebrity T-shirt First

The probability of drawing a celebrity T-shirt first is the number of celebrity T-shirts divided by the total number of T-shirts. This is \(\frac{5}{19}\) since there are 5 celebrity T-shirts out of 19 total.

Step 3: Determine Remaining T-Shirts After First Draw

If a celebrity T-shirt is drawn first, there are now \(19 - 1 = 18\) T-shirts left in the bag.

Step 4: Calculate Probability of Drawing a Yellow T-shirt Second

After one celebrity T-shirt has been drawn, there are still 8 yellow T-shirts out of the remaining 18 T-shirts. Thus, the probability of drawing a yellow T-shirt second is \(\frac{8}{18}\), which simplifies to \(\frac{4}{9}\).

Step 5: Calculate Overall Probability

The overall probability of first drawing a celebrity T-shirt and then a yellow T-shirt is the product of the two individual probabilities: \(\frac{5}{19} \times \frac{4}{9}\). Calculate this to get \(\frac{20}{171}\).

Key Concepts

Independent Events

In probability, independent events are those whose outcomes do not affect each other. Consider tossing a coin and rolling a die. Whether the coin lands on heads or tails does not influence the outcome of the die. These are independent events.
For two events to be independent, the probability of both events occurring should be the product of their individual probabilities. Mathematically, if A and B are two events, they are independent if:

• \(P(A \cap B) = P(A) \times P(B)\).

In our carnival game problem, each draw from the bag affects the other because the second draw's probabilities change when the first T-shirt is drawn, meaning the events are not independent.

Dependent Events

Dependent events occur when the outcome of one event affects the outcome of another. In the carnival game Anita is playing, drawing a T-shirt affects the subsequent probabilities because the T-shirts are not replaced. After removing a T-shirt, fewer options remain, changing the likelihood of future draws.
In our example, the probability of drawing a celebrity T-shirt first affects the probability of drawing a yellow T-shirt second. After one T-shirt is taken, there are now only 18 T-shirts left in the bag. This changes the probability landscape. Thus, when one event impacts another, they are dependent events.

Conditional Probability

Conditional probability represents the probability of an event occurring given that another event has already occurred. It provides a way to update probabilities based on new information.
In our T-shirt drawing example, Anita's situation is one of conditional probability. The probability of drawing a yellow T-shirt on the second draw is conditional upon having drawn a celebrity T-shirt first. This is different from simple probability since part of the information (the first draw) affects the calculation.
Mathematically, conditional probability of event B given that event A has occurred is expressed as:

• \(P(B|A) = \frac{P(A \cap B)}{P(A)}\)

This emphasizes how the probabilities adjust based on preceding outcomes in scenarios like the one Anita faces in her game.