Question

option b: roll two six - sided dice, defining “success” as “the total rolled is seven or higher.”

• determine the theoretical probability of success for your chosen option. (if selecting option c, you may have to do a web search for this information)
• run your experiment twenty times. record your number of successes.
• compute the empirical probability of success using your results.
• using your theoretical probability, compute the probability of observing two successes in a row.
• using your empirical probability, compute the probability of observing two successes in a row.
• determine two conditional probabilities for your option:

o option a: suppose that the first ten cards were revealed to all be black (either a spade or a club). what is the probability that the next card will be red? repeat this question but with the first ten cards all being red.

• make a conclusion.

o when you repeated the experiment 20 times, should we consider each trial dependent or independent from each other?

Explanation:

Step1: Find total outcomes

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

Step2: Find favorable outcomes for theoretical probability

The pairs of dice rolls that result in a sum of 7 or higher are:
Sum of 7: $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$ - 6 pairs
Sum of 8: $(2,6),(3,5),(4,4),(5,3),(6,2)$ - 5 pairs
Sum of 9: $(3,6),(4,5),(5,4),(6,3)$ - 4 pairs
Sum of 10: $(4,6),(5,5),(6,4)$ - 3 pairs
Sum of 11: $(5,6),(6,5)$ - 2 pairs
Sum of 12: $(6,6)$ - 1 pair
The total number of favorable outcomes is $6 + 5+4 + 3+2 + 1=21$.
The theoretical probability $P_{theo}$ of success (sum is 7 or higher) is $\frac{21}{36}=\frac{7}{12}$.

Step3: Empirical probability after 20 trials

Let's assume after 20 trials, the number of successes is $n$. The empirical probability $P_{emp}$ is $\frac{n}{20}$.

Step4: Probability of two successes in a row using theoretical probability

Since the trials are independent, the probability of two successes in a row using the theoretical probability is $P_{theo}\times P_{theo}=(\frac{7}{12})^2=\frac{49}{144}$.

Step5: Probability of two successes in a row using empirical probability

The probability of two successes in a row using the empirical probability is $P_{emp}\times P_{emp}=(\frac{n}{20})^2$.

Step6: Conditional probabilities (not related to Option B, but for general understanding)

In a standard deck of 52 cards, if the first ten cards are black, there are $52 - 10=42$ cards left, and 26 of them are red. So the probability that the next card is red is $\frac{26}{42}=\frac{13}{21}$. If the first ten cards are red, there are $52 - 10 = 42$ cards left, and 26 are black. So the probability that the next card is red is $\frac{26 - 10}{42}=\frac{16}{42}=\frac{8}{21}$.

Step7: Conclusion on independence

When rolling two dice 20 times, each trial is independent because the outcome of one roll does not affect the outcome of another roll.

Answer:

Theoretical probability of success: $\frac{7}{12}$
Empirical probability depends on the number of successes $n$ in 20 trials: $\frac{n}{20}$
Probability of two successes in a row using theoretical probability: $\frac{49}{144}$
Probability of two successes in a row using empirical probability: $(\frac{n}{20})^2$
Conditional probabilities as calculated above
Each trial in the 20 - roll experiment of two dice is independent.