Question

2 a) what is the theoretical probability of rolling a sum of 7 with two standard six - sided dice? b) pascale wanted to compare the theoretical probability to results of an experiment. she rolled 2 dice 50 times, and got a sum of 7 a total of 5 times. calculate the experimental probability. c) if the dice were rolled 1000 times, how many times do you think a sum of 7 would be rolled? would you use theoretical or experimental probability? explain your answer. 3. madeleine is choosing fruit from a fruit bowl. the bowl contains 3 apples, 2 nectarines, and 1 peach. a) if she randomly chooses a fruit without looking, what is the probability that she will take a peach? b) madeleine does take a peach, but decides she wants a different fruit. she places the peach back in the bowl, then randomly chooses again without looking. what is the probability that she will choose an apple?

Explanation:

Step1: Find total outcomes for two - dice roll

When rolling two six - sided dice, the total number of outcomes is $6\times6 = 36$ since each die has 6 possible outcomes.

Step2: Find favorable outcomes for sum of 7

The combinations that give a sum of 7 are $(1,6)$, $(2,5)$, $(3,4)$, $(4,3)$, $(5,2)$, $(6,1)$, so there are 6 favorable outcomes.
The theoretical probability $P_{theoretical}$ of rolling a sum of 7 is $\frac{6}{36}=\frac{1}{6}$.

Step3: Calculate experimental probability

Pascale rolled 2 dice 50 times and got a sum of 7, 5 times. The experimental probability $P_{experimental}=\frac{5}{50}=\frac{1}{10}$.

Step4: Predict number of times sum of 7 is rolled in 1000 rolls

Using the theoretical probability, if the dice are rolled $n = 1000$ times, the expected number of times a sum of 7 is rolled is $E=n\times P_{theoretical}=1000\times\frac{1}{6}=\frac{1000}{6}\approx167$ times. We use the theoretical probability because as the number of trials increases, the experimental probability approaches the theoretical probability.

Step5: Probability of choosing a peach from fruit bowl

The fruit bowl has $3 + 2+1=6$ fruits. The probability of choosing a peach is $\frac{1}{6}$.

Step6: Probability of choosing an apple after replacing the peach

After replacing the peach, there are still 6 fruits in the bowl and 3 apples. The probability of choosing an apple is $\frac{3}{6}=\frac{1}{2}$.

Answer:

a) $\frac{1}{6}$
b) $\frac{1}{10}$
c) Approximately 167 times, use theoretical probability.

1. a) $\frac{1}{6}$

b) $\frac{1}{2}$