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Problem of the Week
Problem B and Solution
On-Base Probabilities

Problem

In baseball, OBP stands for On-Base Percentage. It is the number of times the player makes it on base divided by the total number of times they step up to bat, rounded to three decimal places. For \(10\) players in the MLB, suppose we have the following OBP statistics.

Player OBP
Jake Bauers \(0.357\)
Vladimir Guerrero \(0.500\)
Ian Happ \(0.156\)
Kyle Schwarber \(0.278\)
Andrew Vaughn \(0.267\)
Alex Call \(0.857\)
Jarren Duran \(0.091\)
Aaron Judge \(0.581\)
Sal Stewart \(0.600\)
Masataka Yoshida \(0.571\)

  1. Order the given OBP data on the probability line below from least likely to get on base to most likely to get on base.

    A number line from 0 to 1. Five evenly spaced tick marks divide the line into four intervals. The tick marks, from left to right, are labelled: Impossible (at 0), Unlikely, Even chance, Likely, and Certain (at 1).

  2. Choose any three of the ten players and answer the questions that follow. If they step up to bat \(10\) times, how many times would you expect each of these three players to get on base? What about \(20\) times at bat? \(25\) times?

  3. What would be a reasonable explanation for the OBP of Jasson Dominguez, a brand new player with an incredible OBP of \(1.000\)?

  4. A new player has an OBP of \(0.400\) after \(10\) times at bat, but wants to increase their OBP to \(0.550\). How many times must they make it on base in their next \(10\) times at bat?

Solution

  1. The given OBP data is shown in order on the probability line below. Jarren Duran is least likely to get on base and Alex Call is most likely.

    From left to right, 0.091 and 0.156 are between Impossible and Unlikely; 0.267, 0.278, and 0.357 are between Unlikely and Even chance; 0.500 is at Even chance; 0.571, 0.581, and 0.600 are between Even chance and Likely; and 0.857 is between Likely and Certain.

  2. If a player steps up to bat \(n\) times, the number of times you would expect them to make it on base is equal to \(n\) multiplied by their OBP, rounded to the nearest whole number. The results for \(n=10\), \(20\), and \(25\) times at bat (AB) are in the table for all the players; find your chosen three.

    Player OBP 10 AB 20 AB 25 AB
    Jake Bauers \(0.357\) \(4\) \(7\) \(9\)
    Vladimir Guerrero \(0.500\) \(5\) \(10\) \(13\)
    Ian Happ \(0.156\) \(2\) \(3\) \(4\)
    Kyle Schwarber \(0.278\) \(3\) \(6\) \(7\)
    Andrew Vaughn \(0.267\) \(3\) \(5\) \(7\)
    Alex Call \(0.857\) \(9\) \(17\) \(21\)
    Jarren Duran \(0.091\) \(1\) \(2\) \(2\)
    Aaron Judge \(0.581\) \(6\) \(12\) \(15\)
    Sal Stewart \(0.600\) \(6\) \(12\) \(15\)
    Masataka Yoshida \(0.571\) \(6\) \(11\) \(14\)

  3. Since Jasson Dominguez is a brand new player, his incredible batting average of \(1.000\) likely means that he has had very few times at bat and has made it on base every time.

  4. For an OBP of \(0.55\) after \(20\) times at bat, we need \(\dfrac{n}{20}=0.55\), where \(n\) is the total number of times on base. Thus, \(n=20\times 0.55=11\) times on base. Since the player already has a batting average of \(0.4\) after \(10\) times at bat, they have made it on base \(0.4\times 10=4\) times. Thus, the player needs to make it on base \(7\) more times in their next \(10\) times at bat.