# Probability with cards without replacement

If theres 48 in a deck and 24 of them are red and 24 of them are black. Whats the probability that the next card is red, when 4 out of 12 cards drawn are red without replacement?

Does the success rate(4/12, 33%) influence what the next card drawn will be ? i.e. How would you go about this ?

## 1 Answer

If the deck is shuffled properly, the chance of any card that is drawn will only be affected by how many of each colour are left in the deck. The fact that that you've drawn four red cards only affects the probability in that there are four fewer red cards in the deck.

If you've drawn 12 cards and four of them are red, then 8 are black. Therefore there are 36 cards left, 20 of which are red and 16 of which are black. So the probability of drawing a red card next is $\frac{20}{36} = \frac{5}{9}$.

• Since 5/9 is equal to 55.55% that means i have a +5.55% chance of getting a red card. What % margin is big enough to say i will get a red card? – Rafael Andrade Oct 17 '17 at 16:10
• I'm not sure I understand - are you asking how to guarantee you get a red card? For that you would need every card in the deck to be red. – Prefuzek Oct 17 '17 at 16:33
• lets say you have a 80 % chance of getting a red card. The probability of getting a black card is really low only 20%. But the chances of getting high probability like 80% is less than likely its more like to fall between 40%-60%. So is it safe to say that the probability of getting a red card at 55% and higher will more than likely guarantee me a red card. At a 5% advantage is it less likely the odds will beat it ? – Rafael Andrade Oct 17 '17 at 16:49
• I think you're misunderstanding what probability means. You can't say anything about the likelihood of getting a red card beyond what you've calculated for the probability. If there is a 5/9 chance of drawing a red card, then you're certainly more likely to get a red card then a black card, since 5/9 > 50%. But the only point at which you can say that you WILL get a red card is if the probability is 100%. – Prefuzek Oct 17 '17 at 17:55
• What about finding the probability of defying the odds? – Rafael Andrade Oct 17 '17 at 18:05