## Dependent Events

Dependent events, on the other hand, are outcomes that are affected by other outcomes. Consider the following example.

Example 1

What is the probability of randomly drawing an ace from a deck of cards and then drawing an ace again from the same deck of cards, without returning the first drawn card back to the deck?

For the first draw, the probability of a favorable outcome is , as explained earlier; however, after that first card has been drawn, the total number of outcomes is no longer 52, but now 51 because a card has been removed from the deck. And if that first card drawn resulted in a favorable outcome (an ace), there would now be only three aces in the deck. If that first card drawn were not an ace, the number of favorable outcomes would remain at four. So, the second draw is a dependent event because its probability changes depending upon what happens on the first draw.

If, however, you replace that drawn card back into the deck and shuffle well again before the second draw, then the probability for a favorable outcome for each draw will now be equal , and these events will be independent.

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