Question

events a and b are disjoint. find p(a or b).

1. p(a) = 0.3, p(b) = 0.1 6. p(a) = \frac{2}{3}, p(b) = \frac{1}{5}

problem solving

1. your dart is equally likely to hit any point inside the board

shown. you throw a dart and pop a balloon. what is the
probability that the balloon is red or blue? (see example 1)

Explanation:

Problem 3

Step1: Recall disjoint events formula

For disjoint events \( A \) and \( B \), \( P(A \text{ or } B)=P(A)+P(B) \).

Step2: Substitute values

Given \( P(A) = 0.3 \), \( P(B)=0.1 \), so \( P(A \text{ or } B)=0.3 + 0.1=0.4 \).

Step1: Recall disjoint events formula

For disjoint events \( A \) and \( B \), \( P(A \text{ or } B)=P(A)+P(B) \).

Step2: Substitute and calculate

Given \( P(A)=\frac{2}{3} \), \( P(B)=\frac{1}{5} \). Find a common denominator, which is \( 15 \). Convert fractions: \( \frac{2}{3}=\frac{10}{15} \), \( \frac{1}{5}=\frac{3}{15} \). Then \( P(A \text{ or } B)=\frac{10}{15}+\frac{3}{15}=\frac{13}{15} \).

Step1: Count total and target balloons

First, count total balloons. From the image (assuming the balloons: blue - 1, red - 3, green - 3, yellow - 1? Wait, let's re - examine. Wait, looking at the image: blue (1), red (3), green (4)? Wait, no, let's count again. Wait, the image has: top row: blue, red, green, red; bottom row: green, yellow, red, green. So total balloons: \( 4 + 4=8 \)? Wait, top row 4, bottom row 4, total 8. Blue balloons: 1, red balloons: 3 (top row two reds? Wait, top row: blue, red, green, red (so two reds), bottom row: green, yellow, red, green (one red). So red balloons: \( 2 + 1 = 3 \), blue balloons: 1. So total balloons \( n = 8 \), blue balloons \( n_{blue}=1 \), red balloons \( n_{red}=3 \).

Step2: Use disjoint events formula

Since "red" and "blue" are disjoint events, \( P(\text{red or blue})=P(\text{red})+P(\text{blue}) \). \( P(\text{red})=\frac{3}{8} \), \( P(\text{blue})=\frac{1}{8} \). So \( P(\text{red or blue})=\frac{3 + 1}{8}=\frac{4}{8}=\frac{1}{2} \). Wait, maybe my counting was wrong. Let's count again: top row: blue (1), red (2), green (1), red (1)? No, maybe the correct count: let's list all balloons:

Top row:

1. Blue
2. Red
3. Green
4. Red

Bottom row:

1. Green
2. Yellow
3. Red
4. Green

So red balloons: positions 2,4,7 (3 reds), blue: position 1 (1 blue), green: positions 3,5,8 (3 greens), yellow: position 6 (1 yellow). Total balloons: 8. So number of red or blue balloons: \( 3+1 = 4 \). So probability is \( \frac{4}{8}=\frac{1}{2} \).

Answer:

\( 0.4 \)

Problem 6