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Whenever two or more events happen at the same time, i.e., when we use the word “and,” you will have to decide if the events are independent or dependent.

Examples 6 and 7 are examples of Independent probabilities. The outcome of the first event does not affect the probability of the second. Coin tosses are independent. They cannot affect each other's probabilities; the probability of each toss is independent of a previous toss and will always be 1/2. Separate drawings from a deck of cards are independent events ONLY if you put the cards back.

Dependent events are the opposite. The probability of the second event is affected by the first event. An example of a dependent event is drawing a card from a deck but not returning it. By not returning the card, you've decreased the number of cards in the deck by 1, and you've decreased the number of whatever kind of card you drew. If you draw an ace of spades, then there are 1 fewer aces and 1 fewer spades.

Note: dependent probabilities always coincide with “and” problems, so they will always be multiplication problems.

Example 8
Two cards are pulled from a deck of 52 cards. They are pulled one after the other, and the first is not returned to the deck. What is the probability that both cards will be spades?

Answer
Since both cards must be spades, this is an “and” question. We need to calculate the individual probability of each card and then multiply.

Chapter 4: Working Backwards