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Added solution to extended version of problem
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Philip C
Philip C
  • 119
  • 4

Answer

2/3

Reasoning

In randomly picking a card and randomly picking a side from that card, you have effectively picked a random card-side from all the card-sides in the hat. There are 3 white card-sides in total. 2 of which have another white side on the back. Given that you've picked a white card-side, it follows that there's a 2/3 chance you've picked one of the white card-sides with another white side on the back.

This reasoning extends to any number of cards with any number of colours:

Given that a randomly picked side of a randomly picked card is colour A, what is the probability that the opposite side is colour B?

Count the total number of card-sides of colour A. Call this $N_A$. Then count the number of those card-sides with colour B on the back. Call this $N_{A,B}$. Then the probability of the opposite side being colour B is $N_{A,B}/N_A$.

Answer

2/3

Reasoning

In randomly picking a card and randomly picking a side from that card, you have effectively picked a random card-side from all the card-sides in the hat. There are 3 white card-sides in total. 2 of which have another white side on the back. Given that you've picked a white card-side, it follows that there's a 2/3 chance you've picked one of the white card-sides with another white side on the back.

Answer

2/3

Reasoning

In randomly picking a card and randomly picking a side from that card, you have effectively picked a random card-side from all the card-sides in the hat. There are 3 white card-sides in total. 2 of which have another white side on the back. Given that you've picked a white card-side, it follows that there's a 2/3 chance you've picked one of the white card-sides with another white side on the back.

This reasoning extends to any number of cards with any number of colours:

Given that a randomly picked side of a randomly picked card is colour A, what is the probability that the opposite side is colour B?

Count the total number of card-sides of colour A. Call this $N_A$. Then count the number of those card-sides with colour B on the back. Call this $N_{A,B}$. Then the probability of the opposite side being colour B is $N_{A,B}/N_A$.

Source Link
Philip C
Philip C
  • 119
  • 4

Answer

2/3

Reasoning

In randomly picking a card and randomly picking a side from that card, you have effectively picked a random card-side from all the card-sides in the hat. There are 3 white card-sides in total. 2 of which have another white side on the back. Given that you've picked a white card-side, it follows that there's a 2/3 chance you've picked one of the white card-sides with another white side on the back.