Dependent events means that the outcome of one event affects the outcome of the other event.
Say we pick an ace from a standard deck of 52 cards. We keep the card on our hand, meaning that we don’t put it back in the deck, so the deck is now reduced to 51 cards. In this first draw we had 4 out of 52 (=1/13) probability of picking an ace. Now, for your second draw, we have a 3 out of 51 (=1/17) probability.
Let’s label the two draws from above as events:
- The probability of drawing an ace in the first draw is event A
- The probability of drawing an ace in the second draw is event B
The probability of drawing an ace in the first draw (event A) = 1/13. And because event A has taken place the deck (or the sample space) has been reduced to 51 cards. And because, we draw an ace in the first draw, there are now only 3 aces left in the deck. The probability of event B taking place is therefore 3/51 = 1/17:
P(A) = 1/13
P(B) = 1/17
If we don’t draw an ace in first draw
If we don’t get an ace in the first draw the probability of drawing an ace in the second is still different from the first draw. P(A) is still different from P(B). Because, event A has taken place, the sample space is reduced from 52 to 51, and because event A gave did not return an ace, the 4 aces remain in the deck, so:
P(A) = 1/13
P(B) = 4/51
Drawing 3 marbles example
Say we have 6 green marbles and 4 blue marbles in a bag. You do three random draws from the bag without replacing any marbles to the bag. What is the probability of randomly picking 2 blue and 1 green marbles?
There is 8% probability of event A, B and C to happen.
With or without replacement
The action of putting back the card to the deck or not putting it back, is typically known under the term of replacement. In the example above, we drew a card without replacing it. This is known as an experiment without replacement.
Had we replaced the card to the deck, it would, obviously, be an experiment with replacement. In this case, if we had replaced the card to the deck, the two events would not have been dependent events. The probability of drawing an ace in the second draw would not have been affected by the first draw. P(B) would have been independent of P(A).
Probability distributions like the binomial distribution are conditioned by replacement and that the events are independent.
Dependent events and mutually exclusive events
Mutually exclusive events are dependent event. They cannot happen at the same time. If you flip a head with a coin you cannot flip a tail on that same flip. Whether you flip a head depends on whether you flip a tail, because if you flip tail, you cannot flip head.
From Independent events we recall the comparing independent and mutually exclusive events:
- Khan Academy list of tutorial videos on Probability and combinatorics
- Khan Academy video: Independent & dependent probabilities
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