judging probability by intuition is not always correct?

Question

Suppose I have a question which goes like,"A pack of cards is counted with face downwards, and it is found that one card is missing. Two cards are drawn and are found to be spades. The odds in favor that the missing card is a spade are_______?" I wanted to ask if the odds in favor of the missing card being spade now will be more or less than if we were not mentioned the last piece of information.(regarding that the two cards drawn are spades)?

Please give some reasons in justification of your answer.I feel that the odds now will be less just by intuition.Is intuition in probability always correct?If not can you give me some counter examples.

PS I am sorry if you feel that the second part of the question has no connection with the first part.Also I was not able to find a better title to the question than posted.SORRY!

Answer 1

Statistics is definitely not always intuitive. We use statistics to make inferences and draw general conclusions from limited data. Because of this, we often see patterns within random data sets that cause us to draw unwarranted and fallacious conclusions.

For a very basic example, consider roulette. Many people who bet this game will bet either black or red hoping for a quick payout. People commonly make the mistake of assuming that as a string of occurrences of a single color grows - for example, the ball lands on red three times consecutively - the probability of that color occurring again diminishes. This is an intuitive attempt at statistical reasoning and it is incorrect since these events are actually independent.

For your example, it is now less likely that the missing card is a spade. this is because we know now that there are at most 11 spades remaining either in the deck or missing, whereas there were previously 13 spades unaccounted for. If you were to continue this trend, revealing spades as you turn each new card, you would continue to gather data and be able to reasonably determine that it is less and less likely that the missing card is a spade.

Answer 2

It is now less likely that the missing card is a spade. If you bet that the missing card is a spade, then you are saying that one of the 13 spades is missing. However, you get it confirmed that 2 of them are in fact in the deck. That means that only 11 of the spades could possibly be missing, but any of the 13 hearts, or any of the 13 clubs, or any of the 13 diamonds. Therefore it is more likely that the car is one of the ones with 13 possibilities, than one with only 11.

Look up and compare with the Monty Hall problem. It's a nice example of how intuition can be wrong. Other reasons why intuition can be wrong is the Gambler's Fallacy, for example. We are generally terrible at estimating probabilities.

Answer 3

There are now $50$ possibilities for the missing card; $11$ of these possibilities are spades. So the probability of the missing card being a spade (given the stated information) is $\frac{11}{50}=.22$, as compared to the probability of $.25$ without the given information.