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Examples of Generalized AND Rule

The content delves into the application of the generalized AND rule in probability, specifically focusing on scenarios involving selection without replacement, demonstrating how earlier choices affect the probabilities of subsequent selections.
  • The generalized AND rule is crucial for calculating probabilities where events are not independent, particularly in selection without replacement scenarios.
  • Examples provided include calculating the probability of selecting green balls from a box and picking hearts from a deck of cards, both without replacement.
  • The importance of conditional probabilities is highlighted, showing how the probability of an event changes as choices are made.
  • Practical advice on simplifying calculations by canceling before multiplying to avoid unnecessary complexity is given.
  • The examples underscore the dynamic nature of probabilities in scenarios where the available choices diminish over time.
Understanding the Generalized AND Rule
Selection Without Replacement Explained
Calculating Probabilities with Conditional Probabilities
Practical Calculation Advice

Q: For the second practice problem, how did we get 1/4 as the probability of drawing one heart? Shouldn't it be 13/52?

A: When we draw our first card, there are 13 hearts out of 52 cards. So that's 13/52, which is equivalent to 1/4. 1/4 is the same fraction as 13/52, just in lowest terms.

After we choose a heart, we have one fewer heart in the deck and one fewer total card in the deck. After we choose one heart, we have 51 cards left in the deck, and 12 hearts left, so 12 of 51 cards left are hearts. If we choose another heart, now 11 of 50 cards left are hearts.

So, the probability of 3 hearts in a row is:

13/52 * 12/51 * 11/50 =

1/4 * 4/17 * 11/50

= 1/17 * 11/50 = 11/850

Q: I'm not too familiar with playing cards and the different suits, etc. What are some card basics and do I need to know them for the GRE?

A: There are 52 cards in a standard deck, divided into four "suits" (a "suit" is just a symbol on the card.). There are 13 cards in each suit. The suits are spades, hearts, clubs, diamonds.

Spades and clubs are black. So there are 26 black cards.
Hearts and diamonds are red. So there are 26 red cards.

13 spades (black) + 13 clubs (black) + 13 hearts (red) + 13 diamonds (red) = 52 cards total!

You don't need to memorize this for the GRE, because a question should give you all the background info you need. But I might recommend noting this information in case you see card-related practice problems!