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a die is rolled. find the probability of each outcome. 1. p(less than 3…

Question

a die is rolled. find the probability of each outcome. 1. p(less than 3) 2. p(even) 3. p(greater than 2) 4. p(prime) 5. p(4 or 2) 6. p(integer) a jar contains 65 pennies, 27 nickels, 30 dimes, and 18 quarters. a coin is randomly selected from the jar. find each probability. 7. p(penny) 8. p(quarter) 9. p(not dime) 10. p(penny or dime) 11. p(value greater than $0.15) 12. p(not nickel) 13. p(nickel or quarter) 14. p(value less than $0.20) presentations the students in a class are randomly drawing cards numbered 1 through 28 from a hat to determine the order in which they will give their presentations. find each probability. 15. p(13) 16. p(1 or 28) 17. p(less than 14) 18. p(not 1) 19. p(not 2 or 17) 20. p(greater than 16) the table shows the results of an experiment in which three coins were tossed. outcome: hhh hht hth thh tth tht htt ttt tally: 卌 卌 卌 卌 卌 卌 卌 卌 frequency: 5 5 6 6 7 5 8 8 21. what is the experimental probability that all three of the coins will be heads? what is the theoretical probability? 22. what is the experimental probability that at least two of the coins will be heads? what is the theoretical probability? 23. what is the experimental probability that exactly two of the coins will be tails? what is the theoretical probability? lesson 0 - 3 simple probability

Explanation:

Step1: Recall probability formula

The probability formula is $P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$.

1. $P(\text{less than 3})$ when rolling a die
  • Total number of outcomes when rolling a die is $n = 6$ (the numbers 1, 2, 3, 4, 5, 6).
  • Favorable outcomes (less than 3) are 1 and 2, so $m=2$.
  • $P(\text{less than 3})=\frac{2}{6}=\frac{1}{3}$
2. $P(\text{even})$ when rolling a die
  • Even - numbered outcomes on a die are 2, 4, 6. So $m = 3$.
  • $P(\text{even})=\frac{3}{6}=\frac{1}{2}$
3. $P(\text{greater than 2})$ when rolling a die
  • Outcomes greater than 2 are 3, 4, 5, 6. So $m = 4$.
  • $P(\text{greater than 2})=\frac{4}{6}=\frac{2}{3}$
4. $P(\text{prime})$ when rolling a die
  • Prime - numbered outcomes on a die are 2, 3, 5. So $m = 3$.
  • $P(\text{prime})=\frac{3}{6}=\frac{1}{2}$
5. $P(4\text{ or }2)$ when rolling a die
  • Favorable outcomes are 2 and 4. So $m = 2$.
  • $P(4\text{ or }2)=\frac{2}{6}=\frac{1}{3}$
6. $P(\text{integer})$ when rolling a die
  • All outcomes on a die are integers. So $m = 6$.
  • $P(\text{integer})=\frac{6}{6}=1$
7. $P(\text{penny})$ from the jar
  • Total number of coins in the jar: $65 + 27+30 + 18=140$.
  • Number of pennies is 65.
  • $P(\text{penny})=\frac{65}{140}=\frac{13}{28}$
8. $P(\text{quarter})$ from the jar
  • Number of quarters is 18.
  • $P(\text{quarter})=\frac{18}{140}=\frac{9}{70}$
9. $P(\text{not dime})$ from the jar
  • Number of non - dimes: $65 + 27+18=110$.
  • $P(\text{not dime})=\frac{110}{140}=\frac{11}{14}$
10. $P(\text{penny or dime})$ from the jar
  • Number of pennies or dimes: $65 + 30=95$.
  • $P(\text{penny or dime})=\frac{95}{140}=\frac{19}{28}$
11. $P(\text{value greater than }\$0.15)$ from the jar
  • Quarters have a value greater than $\$0.15$. Number of quarters is 18.
  • $P(\text{value greater than }\$0.15)=\frac{18}{140}=\frac{9}{70}$
12. $P(\text{not nickel})$ from the jar
  • Number of non - nickels: $65+30 + 18=113$.
  • $P(\text{not nickel})=\frac{113}{140}$
13. $P(\text{nickel or quarter})$ from the jar
  • Number of nickels or quarters: $27+18 = 45$.
  • $P(\text{nickel or quarter})=\frac{45}{140}=\frac{9}{28}$
14. $P(\text{value less than }\$0.20)$ from the jar
  • Pennies, nickels, and dimes have a value less than $\$0.20$. Number of such coins: $65+27 + 30=122$.
  • $P(\text{value less than }\$0.20)=\frac{122}{140}=\frac{61}{70}$
15. $P(13)$ when drawing cards numbered 1 - 28
  • Total number of cards is 28.
  • Favorable outcome is 1.
  • $P(13)=\frac{1}{28}$
16. $P(1\text{ or }28)$ when drawing cards numbered 1 - 28
  • Favorable outcomes are 2.
  • $P(1\text{ or }28)=\frac{2}{28}=\frac{1}{14}$
17. $P(\text{less than }14)$ when drawing cards numbered 1 - 28
  • Favorable outcomes are 13.
  • $P(\text{less than }14)=\frac{13}{28}$
18. $P(\text{not }1)$ when drawing cards numbered 1 - 28
  • Favorable outcomes are 27.
  • $P(\text{not }1)=\frac{27}{28}$
19. $P(\text{not }2\text{ or }17)$ when drawing cards numbered 1 - 28
  • Favorable outcomes are $28 - 2=26$.
  • $P(\text{not }2\text{ or }17)=\frac{26}{28}=\frac{13}{14}$
20. $P(\text{greater than }16)$ when drawing cards numbered 1 - 28
  • Favorable outcomes are $28 - 16=12$.
  • $P(\text{greater than }16)=\frac{12}{28}=\frac{3}{7}$
21. Experimental and theoretical probability of all three coins being heads
  • Experimental probability:
  • Total number of trials: $5 + 5+6+6+7+5+8+8=50$.
  • Frequency of getting HHH is 5.
  • Experimental $P(\text{HHH})=\frac{5}{50}=\frac{1}{10}$
  • Theoretical probability: Each coin has…

Answer:

  1. $\frac{1}{3}$
  2. $\frac{1}{2}$
  3. $\frac{2}{3}$
  4. $\frac{1}{2}$
  5. $\frac{1}{3}$
  6. $1$
  7. $\frac{13}{28}$
  8. $\frac{9}{70}$
  9. $\frac{11}{14}$
  10. $\frac{19}{28}$
  11. $\frac{9}{70}$
  12. $\frac{113}{140}$
  13. $\frac{9}{28}$
  14. $\frac{61}{70}$
  15. $\frac{1}{28}$
  16. $\frac{1}{14}$
  17. $\frac{13}{28}$
  18. $\frac{27}{28}$
  19. $\frac{13}{14}$
  20. $\frac{3}{7}$
  21. Experimental: $\frac{1}{10}$, Theoretical: $\frac{1}{8}$
  22. Experimental: $\frac{11}{25}$, Theoretical: $\frac{1}{2}$
  23. Experimental: $\frac{2}{5}$, Theoretical: $\frac{3}{8}$