Al. Category: Homework Fart (c) Find the probability that his shirt number is from 34 to 66 AND he weighs at least 291 pounds. Find the probability that his shirt number is from 34 to 66 OR he weighs at least 291 pounds. Part (e) Find the probability that his shirt number is from 34 to 66 GIVEN that he weighs at least 291 pounds.

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To solve the probability questions you've presented, we'll need to define some events and use fundamental probability formulas. However, since specific numerical data (like the total number of shirt numbers, the distribution of weights, etc.) isn't provided, I'll outline the general approach and formulas you can use to find the answers once you have the necessary information.

### Definitions:
- **Event A**: The player's shirt number is between 34 and 66.
- **Event B**: The player weighs at least 291 pounds.

### a) Probability that his shirt number is from 34 to 66 **AND** he weighs at least 291 pounds (P(A ∧ B)):

This is the probability of both events A **and** B occurring simultaneously.

**Formula:**
$$ P(A \ \text{and} \ B) = P(A \cap B) $$

**Interpretation:**
- If you have a set of players, count how many have shirt numbers between 34 and 66 **and** weigh at least 291 pounds.
- Divide that count by the total number of players to get the probability.

### b) Probability that his shirt number is from 34 to 66 **OR** he weighs at least 291 pounds (P(A ∨ B)):

This is the probability of either event A **or** event B occurring, or both.

**Formula:**
$$ P(A \ \text{or} \ B) = P(A) + P(B) - P(A \cap B) $$

**Interpretation:**
- Calculate the probability of A occurring.
- Calculate the probability of B occurring.
- Subtract the probability of both A and B occurring together to avoid double-counting.

### c) Probability that his shirt number is from 34 to 66 **given** that he weighs at least 291 pounds (P(A | B)):

This is a conditional probability, representing the probability of event A occurring **given** that event B has already occurred.

**Formula:**
$$ P(A | B) = \frac{P(A \cap B)}{P(B)} $$

**Interpretation:**
- Determine the probability of both A and B occurring together.
- Divide by the probability of B occurring to find the likelihood of A under the condition that B has occurred.

### Example (Using Hypothetical Numbers):

Suppose you have the following data:
- Total number of players: 100
- Number of players with shirt numbers from 34 to 66: 30
- Number of players weighing at least 291 pounds: 20
- Number of players who both have shirt numbers from 34 to 66 **and** weigh at least 291 pounds: 5

**a) P(A ∩ B):**
$$ P(A \cap B) = \frac{5}{100} = 0.05 \ (5\%) $$

**b) P(A ∨ B):**
$$ P(A \cup B) = \frac{30}{100} + \frac{20}{100} - \frac{5}{100} = 0.45 \ (45\%) $$

**c) P(A | B):**
$$ P(A | B) = \frac{5}{20} = 0.25 \ (25\%) $$

### Conclusion:

To provide precise numerical answers, you'll need the specific data related to the shirt numbers and weights of the players in question. Once you have that information, you can apply the formulas above to calculate the desired probabilities.
To find the probabilities: - **a)** The probability that a player's shirt number is between 34 and 66 **and** they weigh at least 291 pounds is calculated by dividing the number of players meeting both conditions by the total number of players. - **b)** The probability that a player's shirt number is between 34 and 66 **or** they weigh at least 291 pounds is found by adding the probabilities of each event and subtracting the probability of both events occurring together. - **c)** The probability that a player's shirt number is between 34 and 66 **given** they weigh at least 291 pounds is the probability of both events occurring divided by the probability of the weight condition. For exact numbers, you need specific data on the total number of players and how many meet each condition.

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