Question

1. mrs. williams has a prize box with different colored tickets as shown below. each ticket results in a different type of prize.

3 green tickets
4 yellow tickets
3 purple tickets
pedro will randomly select a ticket, replace it, and then select another ticket. what is the probability that he chooses a yellow ticket and then a purple ticket?
a. 3/25 b. 2/15 c. 4/25 d. 7/20

Explanation:

Step1: Find total number of tickets

Total tickets = Green + Yellow + Purple = \( 3 + 4 + 3 = 10 \)

Step2: Probability of yellow ticket

Probability of yellow (\( P(Y) \)) = \( \frac{\text{Yellow tickets}}{\text{Total tickets}} = \frac{4}{10} \)

Step3: Probability of purple ticket

Probability of purple (\( P(P) \)) = \( \frac{\text{Purple tickets}}{\text{Total tickets}} = \frac{3}{10} \)

Step4: Probability of both events (with replacement)

Since events are independent (replacement), \( P(Y \text{ then } P) = P(Y) \times P(P) = \frac{4}{10} \times \frac{3}{10} = \frac{12}{100} = \frac{3}{25} \) (Wait, no—wait, \( \frac{4}{10} \times \frac{3}{10} = \frac{12}{100} = \frac{3}{25} \)? Wait, no, 12/100 simplifies to 3/25? Wait, 12 ÷ 4 = 3, 100 ÷ 4 = 25? No, 12 and 100 have a common factor of 4? 12 ÷ 4 = 3, 100 ÷ 4 = 25. Wait, but let's recalculate: 4/10 3/10 = 12/100 = 3/25. Wait, but option c is 4/25? Wait, no, maybe I made a mistake. Wait, total tickets: 3 + 4 + 3 = 10? Wait, 3 green, 4 yellow, 3 purple: 3+4=7, 7+3=10. Correct. Yellow is 4, so 4/10. Purple is 3, so 3/10. Multiply: (4/10)(3/10)=12/100=3/25. But option a is 3/25, option c is 4/25. Wait, maybe I miscalculated total tickets? Wait, 3 + 4 + 3 = 10? Wait, 3+4=7, 7+3=10. Yes. Wait, maybe the problem is without replacement? No, it says "replace it". Wait, no—wait, maybe the total is 3+4+3=10? Wait, no, 3+4+3=10? Wait, 3 green, 4 yellow, 3 purple: 3+4+3=10. So yellow is 4, purple is 3. So (4/10)*(3/10)=12/100=3/25. But option a is 3/25. Wait, but let's check the options again. Wait, the options are:

a. 3/25

b. 2/15

c. 4/25

d. 7/20

Wait, maybe I made a mistake in total tickets. Wait, 3 green, 4 yellow, 3 purple: 3+4+3=10? Wait, 3+4=7, 7+3=10. Yes. So 4/10 * 3/10 = 12/100 = 3/25. So the correct answer is a. 3/25. Wait, but let's re-express:

Wait, 4/10 is 2/5, 3/10 is 3/10. 2/5 * 3/10 = 6/50 = 3/25. Yes. So the answer is a. 3/25.

Wait, but earlier I thought maybe c, but no. So step-by-step:

1. Total tickets: 3 + 4 + 3 = 10.

1. Probability of yellow: 4/10 = 2/5.

1. Probability of purple: 3/10.

1. Since replacement, independent events: multiply probabilities: (2/5) * (3/10) = 6/50 = 3/25.

So the correct option is a. 3/25.

Answer:

a. $\frac{3}{25}$