The Idea of Probability

I toss a penny and observe whether it lands heads up or tails up. Suppose the penny is fair, i.e., the probability of heads is 1/2 and the probability of tails is 1/2. This means that

• every occurrence of a head must be balanced by a tail in one of the next two or three tosses.

• if I flip the coin 10 times, it would be almost impossible to obtain 7 heads and 3 tails.

• if I flip the coin many, many times the proportion of heads will be approximately 1/2, and this proportion will tend to get closer and closer to 1/2 as the number of tosses increases.

• regardless of the number of flips, half will be heads and half tails.

• all of the above.

There are two games involving flipping a fair coin. In the first game, you win a prize if you can throw between 45 percent and 55 percent heads; in the second game, you win if you can throw more than 60 percent heads. For each game, would you rather flip the coin 30 times or 300 times?

• 30 times for each game

• 300 times for each game

• 30 times for the first game, and 300 for the second

• 300 times for the first game, and 30 for the second

• The outcomes of the games do not depend on the number of flips

If the individual outcomes of a phenomenon are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions, we say the phenomenon is

• random.

• predictable.

• uniform.

• probable.

• normal.

When two coins are tossed, the probability of getting two heads is 0.25. This means that

• of every 100 tosses, exactly 25 will have two heads.

• the odds against two heads are 4 to 1.

• in the long run, the average number of heads is 0.25.

• in the long run two heads will occur on 25% of all tosses.

• if you get two heads on each of the first five tosses of the coins, you are unlikely to get heads the fourth time.

If I toss a fair coin 5000 times

• and I get anything other than 2500 heads, then something is wrong with the way I flip coins.

• the proportion of heads will be close to 0.5

• a run of 10 heads in a row will increase the probability of getting a run of 10 tails in a row.

• the proportion of heads in these tosses is a parameter

• the proportion of heads will be close to 50.

You read in a book on poker that the probability of being dealt three of a kind in a five-card poker hand is 1/50. What does this mean?

• If you deal thousands of poker hands, the fraction of them that contain three of a kind will be very close to 1/50.

• If you deal 50 poker hands, then one of them will contain three of a kind.

• If you deal 10,000 poker hands, then 200 of them will contain three of a kind.

• A probability of 0.02 is somebody’s best guess for a probability of being dealt three of a kind.

• It doesn’t mean anything, because 1/50 is just a number.

A basketball player makes 160 out of 200 free throws. We would estimate the probability that the player makes his next free throw to be

• 0.16.

• 50-50; either he makes it or he doesn’t.

• 0.80.

• 1.2.

• 80.

In probability and statistics, a random phenomenon is

• something that is completely unexpected or surprising

• something that has a limited set of outcomes, but when each outcome occurs is completely unpredictable.

• something that appears unpredictable, but each individual outcome can be accurately predicted with appropriate mathematical or computer modeling.

• something that is unpredictable from one occurrence to the next, but over the course of many occurrences follows a predictable pattern

• something whose outcome defies description.

You are playing a board game with some friends that involves rolling two six-sided dice. For eight consecutive rolls, the sum on the dice is 6. Which of the following statements is true?

• Each time you roll another 6, the probability of getting yet another 6 on the next roll goes down.

• Each time you roll another 6, the probability of getting yet another 6 on the next roll goes up.

• You should find another set of dice: eight consecutive 6’s is impossible with fair dice.

• The probability of rolling a 6 on the ninth roll is the same as it was on the first roll.

• None of these statements is true.

A poker player is dealt poor hands for several hours. He decides to bet heavily on the last hand of the evening on the grounds that after many bad hands he is due for a winner.

• He's right, because the winnings have to average out.

• He's wrong, because successive deals are independent of each other.

• He's right, because successive deals are independent of each other.

• He's wrong, because he’s clearly on a “cold streak.”

• Whether he’s right or wrong depends on how many bad hands he’s been dealt so far.