Lotería winner
Probability; Difficulty: medium
Probability of exactly one winner in the fifth round
This week’s Riddler Classic is a conditional probability question. The Classic reproduced:
A thousand people are playing Lotería, also known as Mexican bingo. The game consists of a deck of 54 cards, each with a unique picture. Each player has a board with 16 of the 54 pictures, arranged in a 4-by-4 grid. The boards are randomly generated, such that each board has 16 distinct pictures that are equally likely to be any of the 54.
During the game, one card from the deck is drawn at a time, and anyone whose board includes that card’s picture marks it on their board. A player wins by marking four pictures that form one of four patterns, as exemplified below: any entire row, any entire column, the four corners of the grid and any 2-by-2 square.
After the fourth card has been drawn, there are no winners. What is the probability that there will be exactly one winner when the fifth card is drawn?
*spoiler alert* solution and answer follow below
Solution
Interpreting the question as a conditional probability question i.e. given that there are no winners with the first 4 drawn cards, we want the probability that we have exactly one winner out of the 1000 players on the 5th card.
Let’s consider just one card that you have with 16 distinct pictures.
The deck which is being called out has 54 distinct pictures. There are 54! ways or permutations of the cards being called out.
There are 18 combinations of image positions in your card (4 rows, 4 columns, 1 with four corners, 9 two by two squares), each comprising four pictures, that can result in a win.
The number of ways in which the deck being called can allow for a win in exactly four rounds is: 18 . 4! . 50! (18 possible patterns, times the number of permutations of the 4 images being called, times the number of permutations of the remaining 50 cards in the deck.
Since we are told this card of yours is not a winner in four rounds, for the number of possible deck permutations we have to subtract the above from the total possibilities. This gives us the conditional sample space of equally likely card deck permutations:
54! – 18 . 4! . 50!
Now let’s calculate the deck permutations that will result in your card being a winner on the fifth picture called.
This is just 18 . 4 . 50 . 4! . 49! = 18 . 4 . 4! . 50! (18 possible winning patterns, times the number of ways to choose the non match call among the first four, times the number of ways to choose the non match card, times the permutations of the 4 matches, times the permutation of the remaining 49 cards.
Hence, the probability \(p\) of your card being a winner in the fifth call given that it was not a winner in the first four is:
\[
p = \frac{18 . 4 . 4! . 50!}{54! \ – \ 18 . 4! . 50!} \approx 0.00022768
\]
Consequently, the probability of your card not being a winner on the fifth call is \(1 – p \ \approx 0.9997723 \)
Since each Lotería card has its 16 pictures selected independently out of the 54, calculating the probability of exactly one card winning out of a 1000 is just the binomial probability of one success in one thousand tries:
\[
{1000 \choose 1}p \ . \ (1-p)^{999} \approx \mathbf{{\color{Red} {18.14\%} }}
\]
18.14% is the required probability that the next picture called will result in a single winner among the 1000 players given there has been no winner in the first 4 calls.