## Statistics and Probability: When Students Ask

• Why should I bother learning about samples?

Students should be able to recognize when data have been collected from biased samples and understand that biased data and the information derived from it may not be representative of the population.

• How can you be sure your sample is not biased?

To be sure your sample is not biased and truly reflects the population under consideration, you need to ensure that every subject in the population had an equal chance of being selected. One of the most widely known sampling mistakes occurred in 1948, in polls taken during the Presidential race between Harry Truman and Thomas Dewey. The sample that was taken before the election was biased and predicted that Dewey would win. However, the election results proved the prediction inaccurate when Truman was elected President. The sample that was used in the poll came from a subset of people who owned telephones and thus did not reflect the total voting population, since many voters then did not own telephones.

• Why doesn't the theoretical probability of an event always equal the experimental probability of that event?

The theoretical probability of an event is based on logical analysis and serves as the best prediction for what may occur. If you were to flip a coin 100 times, you would expect to get 50 heads and 50 tails. However, you should not be surprised to get 46 heads and 54 tails, or a similar outcome. The fact that reality does not always mirror what should happen theoretically is what probability is all about. The probability of flipping a coin 5 times and getting 5 heads is . So if it does happen, it is an unlikely outcome, but an unlikely outcome is still possible.

• Can an event have more than one sample space?

No, an event can have only one sample space. Let's consider the event of flipping a coin three times. To list the sample space, we find all the possible outcomes as indicated below.

 HHH HHT HTH THH TTH THT HTT TTT

In this sample space each of the outcomes is equally likely, and we can determine from it that the probability of the event of tossing exactly 2 heads is 3/8. You might think that you could also list a sample space of flipping the coin three times as shown below.