1.If two balanced die are rolled, the possible outcomes can be represented as follows.

(1, 1) (21) (3, 1) (4.1) (5,1) (61)

(1,2) (2, 2) (32) (4, 2) (5,2) (6.2)

(1.3) (2, 3) (3, 3) (4,3) (53) (63)

(1,4) (24) (4) (44)(54)64)

(1,5) (2,5) (3,5) (4,5) (5,5) (6.5)

(1, 6) (2,6) (36) (4,6) (5, 6) (6.6)

Determine the probability that the sum of the dice is 11.
2.A bag contains 4 red shirts, 6 blue shirts, and 8 black shirts. If a shirt is randomly selected from the bag, what is the probability that it is red?

3. The following contingency table shows the number of customers who bought various brands of digital cameras at Walmart and Best Buy.

Camera Brand

Best Buy

Walmart

Total

Canon

24

9

30

54

Nikon

15

42

51

Sony

33

48

Fuji Film

6

12

18

Other

69

60

129

Total

123

177

300

What is the probability that a randomly selected customer from this sample bought a Sony camera or purchased the camera at Walmart?
4. The of Events A and B represents the number of instances in which either Event A or Event B or both occur together

Let's go through each question in turn and calculate the probabilities as requested.

### 1. Probability that the sum of two dice is 11

When rolling two six-sided dice, each die has 6 possible outcomes (1 through 6). The total number of possible outcomes is \(6 \times 6 = 36\). We need to find outcomes where the sum of the dice equals 11.

The pairs that sum to 11 are:
- (5,6)
- (6,5)

There are 2 outcomes where the sum is 11. Therefore, the probability of getting a sum of 11 is:
\[
\frac{2}{36} = \frac{1}{18}
\]

### 2. Probability of selecting a red shirt from a bag

The bag contains:
- 4 red shirts
- 6 blue shirts
- 8 black shirts

The total number of shirts is:
\[
4 + 6 + 8 = 18
\]

The probability of selecting a red shirt is the ratio of red shirts to the total number of shirts:
\[
\frac{4}{18} = \frac{2}{9}
\]

### 3. Probability of a customer buying a Sony camera or buying from Walmart

Let's calculate the probability of either buying a Sony camera or buying from Walmart using the provided table.

#### Total customers who bought a Sony camera:
- Sony at Best Buy: 33
- Sony at Walmart: 48
\[
33 + 48 = 81
\]

#### Total customers who bought a camera at Walmart:
- Walmart total: 177

#### Customers who bought a Sony camera at Walmart (to avoid double-counting):
\[
48
\]

Using the formula for the probability of \(A\) or \(B\):
\[
P(\text{Sony or Walmart}) = P(\text{Sony}) + P(\text{Walmart}) - P(\text{Sony and Walmart})
\]
\[
= \frac{81}{300} + \frac{177}{300} - \frac{48}{300} = \frac{210}{300} = \frac{7}{10}
\]

So, the probability that a randomly selected customer bought a Sony camera or purchased from Walmart is \(\frac{7}{10}\).

### 4. Union of Events A and B

The union of Events A and B, denoted \( A \cup B \), represents the occurrence of either Event A or Event B or both. If we let \(n(A)\) and \(n(B)\) represent the number of instances in which Event A and Event B occur, respectively, and \(n(A \cap B)\) represent the number of instances where both occur, then:
\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]

This formula ensures we don’t double-count instances where both events occur.

---

Let me know if you would like further explanation or have questions about any part.

### Further Questions:
1. How would you find the probability of getting a sum of 7 when rolling two dice?
2. What is the probability of selecting a blue shirt from the bag in Question 2?
3. How would the probability change in Question 3 if the total customers were different?
4. How do we calculate the probability of both events \(A\) and \(B\) occurring simultaneously?
5. What is the difference between the union and intersection of two events?

**Tip:** When calculating probabilities with overlapping events, always remember to subtract the probability of their intersection to avoid double-counting.

1. If two balanced dice are rolled, determine the probability that the sum of the dice is 11. 2. A bag contains 4 red shirts, 6 blue shirts, and 8 black shirts. If a shirt is randomly selected from the bag, what is the probability that it is red? 3. What is the probability that a randomly selected customer bought a Sony camera or purchased the camera at Walmart? 4. Define the union of Events A and B.

Explore probability problems including calculating outcomes of dice rolls, selecting shirts from a bag, and determining probabilities in camera purchases. Learn to apply concepts such as the union of events and the addition rule for probability, with detailed solutions provided. Suitable for students in Grades 9-12.

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