Let's address each question:
1. Probability of Disliking Apple Juice
- Total people surveyed = 150
- People who dislike apple juice = 85
Probability of disliking apple juice: \[ \text{Probability} = \frac{\text{Number of people who dislike}}{\text{Total number of people}} = \frac{85}{150} \] To simplify, divide by 5: \[ \frac{85 \div 5}{150 \div 5} = \frac{17}{30} \] Answer: \( \frac{17}{30} \)
2. Equal chances of letter selection from "MISSISSIPPI"
- The counts of letters are:
- M = 1
- I = 4
- S = 4
- P = 2
Only S and I have equal occurrence (4 each).
Answer: S and I
3. Experimental Probability of Land on 3
- Total rolls = 30
- Rolls that landed on 3 = 10
Experimental Probability: \[ \text{Probability} = \frac{10}{30} = \frac{1}{3} \]
4. Experimental vs. Theoretical Probability of Three Heads
- Experimental occurrence = 40 / 100
- Theoretical probability (3 heads from 3 coins) = \( \frac{1}{8} \)
Experimental Probability: \( \frac{40}{100} = \frac{2}{5} = 0.4 \)
Difference: \[ \text{Difference} = \frac{2}{5} - \frac{1}{8} \] Finding a common denominator (40): \[ \frac{16}{40} - \frac{5}{40} = \frac{11}{40} \]
5. Probability Model for Picking a Card from {2, 4, 6, 10}
Each card can be chosen with equal probability: \[ f(x) = \frac{1}{4}, \text{ where } x = 2, 4, 6, 10 \]
6. Expected Shoppers for Plastic Bags
- Prefer plastic bags = 50% of 150 shoppers: \[ \text{Expected} = 0.50 \times 150 = 75 \] Answer: 75
7. Spinner Expected Consonants
Consonants in the spinner: B, C, D, F, G, H (6 consonants out of 8) Expected spins on consonant in 160 spins: \[ \text{Expected} = 160 \times \frac{6}{8} = 120 \]
8. Long-run Relative Frequency of Tails
Total tosses = 300, Heads = 180, thus: \[ \text{Tails} = 300 - 180 = 120 \] Relative frequency of tails: \[ \text{Probability} = \frac{120}{300} = 0.4 \]
9. Probability Model for Pick from {5, 10, 15}
Equal probabilities for 3 choices: \[ f(x) = \frac{1}{3}, \text{ where } x = 5, 10, 15 \]
10. Value of x in Probability Model
The sum of probabilities must equal 1: \[ 0.25 + 0.18 + 0.30 + 0.17 + x = 1 \ x = 1 - (0.25 + 0.18 + 0.30 + 0.17) = 0.10 \] Answer: 0.10
11. Expected Difference in White and Red Balls in Bowl B
Total balls in Bowl B = 500 Ratio from the table:
- White balls proportion: \( \frac{60}{100} \)
- Red balls proportion: \( \frac{40}{100} \)
Expected white balls: \[ \text{White} = 500 \times \frac{60}{100} = 300 \] Expected red balls: \[ \text{Red} = 500 \times \frac{40}{100} = 200 \] Difference: \[ \text{Difference} = 300 - 200 = 100 \]
12. Probability of Rolling Odd then Even
- Odd on first roll: \( \frac{3}{6} = \frac{1}{2} \)
- Even on second roll: \( \frac{3}{6} = \frac{1}{2} \)
Combined probability: \[ \text{Probability} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]
13. Probability of Selecting Two Carnations
- Total flowers = 5 Begonias + 6 Carnations + 7 Tulips = 18
- Probability of selecting two carnations: \[ P(\text{1st Carnation}) = \frac{6}{18} \ P(\text{2nd Carnation}) = \frac{5}{17} \ \text{Combined} = \frac{6}{18} \times \frac{5}{17} = \frac{30}{306} = \frac{5}{51} \]
14. Probability of Two Diamonds from Deck of Cards
Total diamonds = 13 out of 52 cards:
- Probability for first diamond: \( \frac{13}{52} \)
- For second diamond (without replacement): \( \frac{12}{51} \)
Combined probability: \[ P = \frac{13}{52} \times \frac{12}{51} = \frac{156}{2652} = \frac{1}{17} \] In percent: \( \frac{1}{17} \times 100 \approx 5.88% \)
15. Probability of Pulling Red Marble
Total marbles = 120, Black = 80, White = 28, Red = 120 - (80 + 28) = 12. The probability of pulling a red marble is: \[ P(\text{Red}) = \frac{12}{120} = \frac{1}{10} \] Thus, the probability of Nicole pulling a red marble from the bowl is \( \frac{1}{10} \) or 10%.
Feel free to ask if you need further explanations or details regarding any section!