When they have made several baskets in succession, basketball players are often described as being "hot".  When they have been unsuccessful for a while, they are described as being "cold" or "in a slump".  Fans and basketball players alike tend to believe that players shoot in streaks.  That is, players have long periods when they are shooting better than we would expect followed by long periods when they aren't doing as well as we would expect.  Is it true that basketball players exhibit streaky behavior in shooting?  Let's first examine streaks of heads and tails in coin flips.

 In the following two sequences of 200 coin flips, H stands for head and T for a tail.  One of the sequences is the result from actually tossing a fair coin.  The other is made up.  Which do you think is the real sequence?

THTTTTHHTHTHTTTTTHHTHTHHHHHHHHHTHTHTHHHTHHHTTHHTHTHHHHH THHTHHTTHTHTHTTTTTTHHTHTHTHTTTHTTTTHHTTTTHTTTTHTTTTHHHTHT HHTTTHTTTHTTHHTHHHTHHHHHTHHHTTHHHTHHTTTTTTTHTTTHTTTHTTHT TTHTTHHHHTHHHTTTHTTTTTTHHTHTTTHH

THTHTTTHTTTTHHTHTTTHTTHHHTHHTHTHTHTTTTHHTTHHTTHHHTHHHTTH HHTTTHHHTHHHHTTTHTHTHHHHTHTTHHHTHHTHTTTHHTHHHTHHHHTTHTH HTHHHTTTHTHHHTHHTTTHHHTTTTHHHTHTHHHHTHTTHHTTTTHTHTHTTHTH HTTHTTTHTTTTHHHHTHTHHHTTHHTHHTHH

Class Vote: Which one is real?    Sequence 1________ Sequence 2_______

1. a.  What was the longest run of heads in your own made-up sequence?  _____
    b.  What is the class average of the longest run of heads for the made-up sequences?_____
    c.  What was the longest run of heads in your randomly generated sequence? _____
    d.  What is the class average of the longest run of heads from the randomly generated sequences? _____

2. Using the answers to question #1 as a clue, in what way does your made-up sequence differ from your randomly generated sequence?

3. Would you like to vote again?  Which one of the sequences above is real? ________

4. If a basketball player is a 50% field-goal shooter and shoots 200 times in a series of games, what would you expect the longest streak of baskets to be if the player doesn't exhibit unusually streaky behavior?  How could you tell if the player was hot?  (Hint: What was the class average longest streak in the real sequence?)
 

5. A gambler knows that red and black are equally likely to occur on each spin of a roulette wheel.  He observes five consecutive reds and bets heavily on red at the next spin.  Asked why, he says that "red is hot" and that the run of reds is likely to continue.  Explain to the gambler what is wrong with this reasoning.
 

6. After hearing you explain why red and black remain equally probable after five reds on the roulette wheel, the gambler moves to a poker game.  He is dealt five straight red cards.  He remembers what you said and assumes that the next card dealt in the same hand is equally likely to be red or black.  Is the gambler right or wrong?  Why?

7. The baseball player Tony Gwynn gets a hit about 35% of the time over an entire season.  After he has failed to hit safely in six straight at-bats, the TV commentator says, "Tony is due for a hit by the law of averages."  Is that right?  Why?
 
 

8. Construct the probability distribution for the length of the longest run of heads when a coin is tossed one, two, three, and four times.  For example, when a coin is tossed two times, there are 22 possible outcomes: HH, HT, TH and TT.  The longest runs of heads in these outcomes are: 2, 1, 1, and 0.  The probability distribution for two, five and six  tosses are completed below.  Compete the other probability distributions.

Run of Heads (1 toss)       Probability                                     Run of Heads (2 tosses)       Probability
         0                               ____                                                        0                                    1/4
         1                               ____                                                        1                                    2/4
                                                                                                          2                                    1/4
 

Run of Heads (3 tosses)    Probability                                     Run of Heads (4 tosses)        Probability
         0                               ____                                                        0                                    ____
         1                               ____                                                        1                                    ____
         2                               ____                                                        2                                    ____
         3                               ____                                                        3                                    ____
         4                               ____
 

Run of Heads (5 tosses)      Probability
         0                               1/32
         1                               11/32
         2                               12/32
         3                               5/32
         4                               2/32
         5                               1/32

9. You would hope that there is some wonderful mathematical way of calculating these probabilities, and there is, but I'm not telling.  In the mean time: how probable is it to have streaks of 4 or more heads in five tosses?

10. If Sonnet was a 50% field goal shooter, and she shot 5 baskets in one quarter of a basketball game, how probable is it, based on chance alone (and the results of our tables), that she could make three or more shots in a row?

11.If you were to make histograms of the distributions, what would be noticeable about the center, spread, and symmetry as the number of tosses increases?