Will the lost card affect?

Probability Level 4

One of a pack of 52 cards has been lost. From the remainder of the pack two cards are drawn and are found to be spades. Find the chance that the missing card is a spade.

The answer is 0.22.

2 solutions

Michael Mendrin
Nov 28, 2015

Let’s pick $3$ cards from a standard $52$ card deck. What’s the probability that the $2$ nd and $3$ rd cards drawn are both spades?

$\dfrac { 39\cdot 13\cdot 12 }{ 52\cdot 51\cdot 50 } +\dfrac { 13\cdot 12\cdot 11 }{ 52\cdot 51\cdot 50 } =\dfrac { 1 }{ 17 }$

What’s the probability that all $3$ cards picked are spades?

$\dfrac { 13\cdot 12\cdot 11 }{ 52\cdot 51\cdot 50 } =\dfrac { 11 }{ 850 }$

Hence, the probability that the “missing card” was a spade is that fraction

$\dfrac { \dfrac { 11 }{ 850 } }{ \dfrac { 1 }{ 17 } } =\dfrac { 11 }{ 50 } =0.220$

Without having drawn the $2$ nd and $3$ rd cards, the probability would have been $0.250$ . After drawing the two cards and discovering both to be spades reduces this to $0.220$ .

Interesting! I went with Bayes' Theorem on this one:

$\dfrac{\dfrac{12\cdot11}{51\cdot50}}{\dfrac{12\cdot11}{51\cdot50} + \dfrac{13\cdot12}{51\cdot50}+ \dfrac{13\cdot12}{51\cdot50}+ \dfrac{13\cdot12}{51\cdot50}} = 0.22$

Each additional denominator term is for each other suit that could be missing and the double spade outcome occurred.

Andrew Ellinor - 5 years, 6 months ago

Paulo Filho
May 21, 2016

After the two spades are drawn, we have 49 cards in the pack and 1 missing. In these 50 cards, 11 are spades. Then, the chance of the missing card being a spade is 11/50.

Will the lost card affect?

The answer is 0.22.

2 solutions

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