Numbers for Words People

Don’t trust your intuition. Part 1: How many batting lineups for a cricket team?

In the game of cricket, two teams take turns to bat or bowl. While only some players bowl, all eleven team members take part in batting. The bowling team aims to dismiss batters, one at a time, and the batting team sends out a replacement, in predetermined order, for each dismissed player. The order in which the batters come out is referred to as the “lineup” of the team. How many possible lineups do you think there are? Is your guess a few dozen, a few hundred possible lineups? You’ll be surprised that the answer is in the millions!

Recreational cricket game
Recreational cricket game. Photo by shents on Pxabay.com

Since I cannot list millions of lineups here, I will start with a simple example. We will see how many possible arrangements there are for 3 players and extrapolate from 3 to 11.

A cricket team of 3 players

The Australian men’s cricket team has been shrunk, for the purposes of this blog, to three players: Travis, Jake and Marnus.

Possible lineups starting with Travis:

Travis, Jake, Marnus
Travis, Marnus, Jake

Similarly, we have 2 lineups starting with Jake and another 2 starting with Marnus. This gives a total of 6 possible lineups for a team of 3 batters.

To bat at number 1, we have 3 options. For each of these, we have 2 options remaining for the number 2 spot. Then, there is 1 option left to go in the number 3 spot. Below is an illustration of this.

Picking cricket lineup
Picking first, second and third batters

We will read the “tree diagram” above from left to right, starting at the top. As can be seen, there are 3 possibilities for the first place, each corresponding to 2 more possibilities for the second spot. Only one possibility is left, each time, for the final spot.

Possible lineups=3×2×1=6Possible\ lineups = 3 \times 2 \times 1 = 6

Batting lineup for 11 players

Getting back to 11 players, the number of possible lineups will be

11×10×9×8×7×6×5×4×3×2×1=39,916,80011 \times 10 \times9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 39,916,800

This is a number just short of 40 million possible lineups! The next time you are mad at the selection committee of a cricketing nation, take that into account.

Do you have any other examples of mathematical questions where the answer defies most people’s intuition? I would love for you to leave it in the comments.

Responses

  1. Steven Francis Avatar
    Steven Francis

    The Monty Hall Problem is probably the one that comes first to mind. People just see it as two options, so 50% chance of success.

    Also, the Birthday Paradox, in terms of how many people do you need in a room before there’s a 50% chance two people share the same birthday. It “feels” like it should be over 100.

    The perimeter of a Koch snow flake. “feels” pretty finite.

    1 = 0.99999999… (recurring)… most people “feel” like its only approximately true.

    The Baysian Trap: You tested positive to a disease. The disease effects 0.1% of the population. If you have the disease, there is a 99% chance the test will correctly identify this. If you don’t have the disease, there is a 1% chance the test will incorrectly identify you having the disease. So given you’ve tested positive, what is the probability you actually have the disease? Most people would “feel” like its got to very very likely.

    Reply
    1. Ziad Baroudi Avatar
      Ziad Baroudi

      Thanks for that, Steven. The Monty Hall problem is on the plan for part 2 of “Don’t trust your intuition”. I’ll definitely consider your other suggestions as well.

      Reply
  2. Don't trust your intuition, part 3: Exponential growth – Numbers for Words People

    […] 1: How many cricket lineups? This post explains […]

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