Probability problem -- Drawing cards with different colors on their 2 sides

In summary: Just keep working at it and you too will become very good at this.In summary, the probability that the other side of the card is also green, given that one side is observed to be green, is 2/3. This can be demonstrated by considering all equally likely options and observing that in 2 out of 3 cases, the other side is also green.
  • #1
TheMathNoob
189
4

Homework Statement


A box contains three cards. One card is red on both sides, one card is green on both sides, and one card is red on one side and green on the other. One card is selected from the box at random, and the color on one side is observed. If this side is green, what is the probability that the other side of the card is also green?

Homework Equations


P(BlA)=P(AnB)/P(A)

The Attempt at a Solution


Why intuitively the prob is not 1/2?

If you pick one card and notice the the first side is green then the prob that other side is green too would be 1/2 because there are just 2 cards that have green sides.
 
Physics news on Phys.org
  • #2
TheMathNoob said:

Homework Statement


A box contains three cards. One card is red on both sides, one card is green on both sides, and one card is red on one side and green on the other. One card is selected from the box at random, and the color on one side is observed. If this side is green, what is the probability that the other side of the card is also green?

Homework Equations


P(BlA)=P(AnB)/P(A)

The Attempt at a Solution


Why intuitively the prob is not 1/2?

If you pick one card and notice the the first side is green then the prob that other side is green too would be 1/2 because there are just 2 cards that have green sides.

Why not do the experiment repeatedly and see what happens?

If you are able, you could write a computer program to model this: pick a card at random, pick a side at random, if it's green, then record the colour of the other side.

Or, you could model it simply by going through all the equally likely options: Card 1, side 1; Card 1, side 2 ...
 
  • #3
There are, initially, 6 possible "sides" of these cards: R/R, R/G, G/G. Since each side of R/R or G/G is separate, to be sure of "equally likely" sides, label those R1/R2, R/G, G1/G2. If you look at one side of a card and it is green, you know it is "G of G/R" or "G1 of G1/G2" or "G2 of G2/G1". Of those three equally likely cases, the other side is also green in the "G1 of G1/G2" case or the "G2 of G2/G1" case, 2 out of the three cases.
 
  • #4
HallsofIvy said:
There are, initially, 6 possible "sides" of these cards: R/R, R/G, G/G. Since each side of R/R or G/G is separate, to be sure of "equally likely" sides, label those R1/R2, R/G, G1/G2. If you look at one side of a card and it is green, you know it is "G of G/R" or "G1 of G1/G2" or "G2 of G2/G1". Of those three equally likely cases, the other side is also green in the "G1 of G1/G2" case or the "G2 of G2/G1" case, 2 out of the three cases.
I agree, but I feel your notation is unclear.
You are right that the best way to think of it is as selecting one of six possible sides (rather than selecting a card, then selecting one side of it).
Three sides are red, three are green, so observing green leaves three equally likely possibilities for which side was selected. In only one of these is the other side red.
 
  • #5
haruspex said:
I agree, but I feel your notation is unclear.
You are right that the best way to think of it is as selecting one of six possible sides (rather than selecting a card, then selecting one side of it).
Three sides are red, three are green, so observing green leaves three equally likely possibilities for which side was selected. In only one of these is the other side red.
I see your intelligence and the questions that people in this forum make and I feel like the dumbest guy in this world XD. In my world, I am smart. This forum inspires my personal growth because I thought that I knew a lot, but I know nothing.
 
  • #6
TheMathNoob said:
I see your intelligence and the questions that people in this forum make and I feel like the dumbest guy in this world XD. In my world, I am smart. This forum inspires my personal growth because I thought that I knew a lot, but I know nothing.

You should not be so hard on yourself. Some of your posts are very good, indeed, and some of your solutions are "spot on"---just not this time. Nobody's perfect, and we all have made (and in my case, continue to make) mistakes once in a while.
 

FAQ: Probability problem -- Drawing cards with different colors on their 2 sides

1. What is the probability of drawing a card with a different color on each side?

The probability of drawing a card with different colors on each side depends on the number of cards in the deck and the number of colors represented. For example, if there are 52 cards in the deck and 4 different colors, the probability would be 4/52 or approximately 7.7%.

2. How does the number of colors in the deck affect the probability of drawing a card with different colors on each side?

The more colors represented in the deck, the lower the probability of drawing a card with different colors on each side. This is because the number of possible combinations of colors increases, making it less likely to draw a card with different colors on each side.

3. Is it more likely to draw a card with different colors on each side if the deck is larger?

No, the size of the deck does not necessarily affect the probability of drawing a card with different colors on each side. As long as the number of colors represented stays the same, the probability remains constant regardless of the deck size.

4. Can the probability of drawing a card with different colors on each side be calculated for a deck with an uneven number of colors?

Yes, the probability can still be calculated for a deck with an uneven number of colors. Simply divide the number of colors by the total number of cards in the deck to determine the probability.

5. How can the probability of drawing a card with different colors on each side be affected by shuffling the deck?

Shuffling the deck does not affect the probability of drawing a card with different colors on each side. As long as the deck remains the same and the number of colors represented remains the same, the probability remains constant.

Back
Top