Question

experimental and theoretical probability
in questions 1 - 7, read and answer each of the questions below. use the number bank to help you check your solutions. not all numbers will be used.

1. in the media - cabinet of jacquelyns house, there are 7 comedy dvds, 4 mystery dvds, and 4 documentary dvds. what is the probability of randomly selecting a mystery dvd from the cabinet?
2. ms. mitchells coin purse has 20 coins. there are 6 pennies, 4 quarters, 3 dimes, and the remainder are nickels. what is the theoretical probability of randomly selecting a nickel from ms. mitchells coin purse?
3. the spinner below is spun 10 times. if the experimental probability of landing on a 3 is $\frac{1}{2}$, then what is the difference between the experimental and the theoretical probabilities?
4. the following cards are used in a game. if each of the cards is turned over and shuffled, then how many times more likely is the chance of drawing a spade over drawing a 7?
5. a fair coin is tossed in the air four times. if the experimental probability of landing on tails is $\frac{1}{2}$, then what is the difference between the experimental and theoretical probability?
6. during a team - building game, participants reach into a bag and randomly select a colored flag, which determines their team. if there are 7 green flags, 5 red flags, and 3 yellow flags, then what is the theoretical probability of selecting a red flag?
7. a wedding planner creates a table to keep track of the meal that each guest selected. which of the following statements is not true?

meal	quantity
chicken	54
seafood	27

a. a guest was half as likely to choose seafood than chicken.
b. a guest was more likely to choose chicken over vegetarian or seafood.
c. a guest was three times as likely to choose vegetarian than chicken.

Explanation:

Step1: Calculate total number of DVDs

Total DVDs = 7 (comedy) + 4 (mystery) + 4 (documentary)=15.

Step2: Calculate probability of selecting mystery DVD

Probability = $\frac{\text{Number of mystery DVDs}}{\text{Total number of DVDs}}=\frac{4}{15}$.

Step3: Calculate number of nickels in coin - purse

Number of nickels = 20-(6 + 4+3)=7.

Step4: Calculate probability of selecting a nickel

Probability = $\frac{\text{Number of nickels}}{\text{Total number of coins}}=\frac{7}{20}$.

Step5: Calculate theoretical probability of spinner landing on 3

Spinner has 4 equal - sized sections, so theoretical probability of landing on 3 is $\frac{1}{4}$. Experimental probability is $\frac{1}{2}$.

Step6: Calculate difference between experimental and theoretical probabilities

Difference = $\frac{1}{2}-\frac{1}{4}=\frac{2 - 1}{4}=\frac{1}{4}$.

Step7: Count number of spades and 7s in cards

There are 2 spades and 1 seven. Probability of drawing a spade is $\frac{2}{6}=\frac{1}{3}$, probability of drawing a 7 is $\frac{1}{6}$.

Step8: Calculate how many times more likely to draw a spade than a 7

$\frac{\text{Probability of drawing a spade}}{\text{Probability of drawing a 7}}=\frac{\frac{1}{3}}{\frac{1}{6}} = 2$.

Step9: Calculate theoretical probability of coin landing on tails

For a fair coin, theoretical probability of landing on tails is $\frac{1}{2}$. Experimental probability is $\frac{1}{4}$.

Step10: Calculate difference between experimental and theoretical probabilities

Difference = $\frac{1}{2}-\frac{1}{4}=\frac{1}{4}$.

Step11: Calculate total number of flags

Total flags = 7 (green)+5 (red)+3 (yellow)=15.

Step12: Calculate probability of selecting a red flag

Probability = $\frac{\text{Number of red flags}}{\text{Total number of flags}}=\frac{5}{15}=\frac{1}{3}$.

Step13: Analyze meal - selection probabilities

Total number of guests = 18 (vegetarian)+54 (chicken)+27 (seafood)=99.
Probability of choosing seafood = $\frac{27}{99}=\frac{3}{11}$, probability of choosing chicken = $\frac{54}{99}=\frac{6}{11}$, probability of choosing vegetarian = $\frac{18}{99}=\frac{2}{11}$.
A guest is half as likely to choose seafood than chicken since $\frac{3}{11}=\frac{1}{2}\times\frac{6}{11}$. A guest is more likely to choose chicken over vegetarian or seafood. A guest is not three times as likely to choose vegetarian than chicken since $\frac{2}{11}
eq3\times\frac{6}{11}$.

Answer:

1. $\frac{4}{15}$
2. $\frac{7}{20}$
3. $\frac{1}{4}$
4. 2
5. $\frac{1}{4}$
6. $\frac{1}{3}$
7. C. A guest was three times as likely to choose vegetarian than chicken.