Basketball Scores

A basketball player takes 100 free throws. She makes the first and misses the second. From the third throw on, the chance she makes a throw equals the fraction of her throws so far that she has made. What is the probability she ends with exactly 50 makes?

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#The rule is a Polya urn

After two throws she sits at one make and one miss. Put one red ball (a make) and one blue ball (a miss) in an urn. Drawing a ball, noting its colour, and returning it with one more of the same colour reproduces the rule exactly. After $k$ throws the urn holds $k$ balls, $M_k$ of them red, and the next throw is a make with probability $M_k / k$ , the fraction made so far. This is the standard Polya urn.

#Uniform by induction

The claim is that for every $n \ge 2$ , the number of makes $M_n$ is uniform on $\{1, 2, \dots, n-1\}$ .

The base case is $n = 2$ , where $M_2 = 1$ with certainty and $\{1,\dots,n-1\} = \{1\}$ . For the step, assume $M_n$ is uniform on $\{1,\dots,n-1\}$ , so each value has probability $\tfrac{1}{n-1}$ . The next throw lifts the count by one with probability $M_n/n$ and leaves it with probability $1 - M_n/n$ , so for any $j$ ,

\PP(M_{n+1} = j) = \PP(M_n = j)\Big(1 - \tfrac{j}{n}\Big) + \PP(M_n = j-1)\,\tfrac{j-1}{n}. \tag{1}

For an interior $j \in \{2,\dots,n-1\}$ both terms appear and collapse,

\frac{1}{n-1}\left[\Big(1 - \tfrac{j}{n}\Big) + \tfrac{j-1}{n}\right] = \frac{1}{n-1}\cdot\frac{n-1}{n} = \frac{1}{n}. \tag{2}

At the ends $j = 1$ and $j = n$ only one term survives, and each still gives $\tfrac{1}{n}$ . So $M_{n+1}$ is uniform on $\{1,\dots,n\}$ , closing the induction.

#Uniform over all tallies

At $n = 100$ the makes are uniform on $\{1,\dots,99\}$ , the forced first make and second miss ruling out $0$ and $100$ . Every tally is equally likely, so

\PP(M_{100} = 50) = \frac{1}{99}. \tag{3}

The Polya urn flattens the distribution completely. Every final tally from 1 to 99 is equally likely, so landing on exactly 50 makes carries probability 1/99.

The surprise is that 50 is no more likely than 1 or 99. The reinforcement keeps whatever lead it stumbles into early, smearing the final count flat across its whole range.