Question

A deck of 10 cards, labeled 1 through 10, is shuffled. A card is drawn at random and then shuffled back into the deck. This is repeated two more times. The card labeled with the number 6 is drawn 2 out of the 3 times.

The predicted probability of drawing the card with the number 6 when one card is drawn is 1/10.

Select the statement that best explains why the observed frequency is different than the predicted probability.

a. They most likely differ because the sample size is too small.

b. They most likely differ because the card is drawn more than 1 time.

c. They most likely differ because the predicted probability is inaccurate.

d. They most likely differ because 2/3 and 1/10 have different denominators.

Answer 1

The discrepancy between observed frequency and predicted probability is attributable to the small sample size used in the experiment, which in this case, resulted in drawing the number 6 two times out of three independent draws from a deck of 10 cards.

Step-by-step explanation:

The observed frequency of drawing the card with the number 6 two times out of three attempts is different than the predicted probability due to the small sample size. The predicted probability of drawing any single card in a deck of 10 is 1/10, which is based on the assumption of equally likely outcomes in a sufficiently large number of trials. When the experiment is conducted only three times, there is a significant chance for variability, and the relative frequency could differ substantially from the theoretical probability.

This is an example of random sampling with replacement, as the card is shuffled back into the deck after each draw. The events are independent since the composition of the deck remains unchanged after each draw, maintaining consistent probability across trials. Over a large number of trials, the law of large numbers suggests that the observed frequency would approach the theoretical probability, but with only three trials, significant deviations can occur.