Questions: There is a pack of four cards numbered 1 to 4. There is also a coin with one side marked as heads and the other tails. As a trial of an experiment, a card was drawn and the coin was flipped. The number (1 to 4) of the card and the side (H for heads and T for tails) of the coin from the flip were recorded. Here is a summary of the data from 550 trials. Outcome 1 H 2 H 3 H 4 H 1 T 2 T 3 T 4 T --------------------------- Number of trials 70 71 66 74 63 69 73 64 Answer each part. (a) Assuming the card was chosen at random and the coin is fair, find the theoretical probability of this event: both drawing the 1, 2, or 3 card and flipping heads, in a single trial. Round your answer to the nearest thousandth. (b) Use the data to find the experimental probability of this event: both drawing the 1, 2, or 3 card and flipping heads, in a single trial. Round your answer to the nearest thousandth. (c) Choose the statement that is true. With a large number of trials, there must be no difference between the experimental and theoretical probabilities. With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small. With a large number of trials, there must be a large difference between the experimental and theoretical probabilities.

There is a pack of four cards numbered 1 to 4. There is also a coin with one side marked as heads and the other tails. As a trial of an experiment, a card was drawn and the coin was flipped. The number (1 to 4) of the card and the side (H for heads and T for tails) of the coin from the flip were recorded. Here is a summary of the data from 550 trials. Outcome 1 H 2 H 3 H 4 H 1 T 2 T 3 T 4 T --------------------------- Number of trials 70 71 66 74 63 69 73 64 Answer each part. (a) Assuming the card was chosen at random and the coin is fair, find the theoretical probability of this event: both drawing the 1, 2, or 3 card and flipping heads, in a single trial. Round your answer to the nearest thousandth. (b) Use the data to find the experimental probability of this event: both drawing the 1, 2, or 3 card and flipping heads, in a single trial. Round your answer to the nearest thousandth. (c) Choose the statement that is true. With a large number of trials, there must be no difference between the experimental and theoretical probabilities. With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small. With a large number of trials, there must be a large difference between the experimental and theoretical probabilities.
Transcript text: There is a pack of four cards numbered 1 to 4. There is also a coin with one side marked as heads and the other tails. As a trial of an experiment, a card was drawn and the coin was flipped. The number (1 to 4) of the card and the side ($H$ for heads and $T$ for tails) of the coin from the flip were recorded. Here is a summary of the data from 550 trials. \begin{tabular}{|r|c|c|c|c|c|c|c|c|} \hline Outcome & $1 H$ & $2 H$ & $3 H$ & $4 H$ & $1 T$ & $2 T$ & $3 T$ & $4 T$ \\ \hline Number of trials & 70 & 71 & 66 & 74 & 63 & 69 & 73 & 64 \\ \hline \end{tabular} Answer each part. (a) Assuming the card was chosen at random and the coin is fair, find the theoretical probability of this event: both drawing the 1, 2, or 3 card and flipping heads, in a single trial. Round your answer to the nearest thousandth. $\square$ (b) Use the data to find the experimental probability of this event: both drawing the 1, 2, or 3 card and flipping heads, in a single trial. Round your answer to the nearest thousandth. $\square$ (c) Choose the statement that is true. With a large number of trials, there must be no difference between the experimental and theoretical probabilities. With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small. With a large number of trials, there must be a large difference between the experimental and theoretical probabilities.
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Solution

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Solution Steps

Step 1: Find the theoretical probability of drawing a 1, 2, or 3 card.

There are 4 cards in total, numbered 1 through 4. The cards numbered 1, 2, or 3 represent $\frac{3}{4}$ of the possible outcomes.

Step 2: Find the theoretical probability of flipping heads.

Since the coin is fair, the probability of flipping heads is $\frac{1}{2}$.

Step 3: Find the theoretical probability of both events occurring.

Since the card draw and coin flip are independent events, we can multiply their probabilities together to find the probability of both occurring. $P(\text{1, 2, or 3 card and heads}) = P(\text{1, 2, or 3 card}) \times P(\text{heads}) = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8}$

Step 4: Convert the fraction to a decimal rounded to the nearest thousandth.

$\frac{3}{8} = 0.375$

Step 5: Find the experimental probability of drawing a 1, 2, or 3 card and flipping heads.

From the data, the number of trials with outcome 1H, 2H, or 3H is $70 + 71 + 66 = 207$. The total number of trials is 550. The experimental probability is:

$P(\text{1, 2, or 3 and heads}) = \frac{207}{550} \approx 0.37636 \approx 0.376$

Step 6: Determine which statement is true.

With a large number of trials, the experimental probability should get closer to the theoretical probability. There might be a small difference between the experimental and theoretical probabilities.

Final Answer:

(a) 0.375 (b) 0.376 (c) With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small.

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