Dependent Events and Sample Space

I. Intro 

Dependent events are affected by the outcome of preceding events. In other words, two events A and B, are dependent if the occurrence of A affects the probability of B. 

I. Dependent Events

A. Using the Multiplication Rule for Dependent Events

Example 1

What's the probability of drawing two sevens from a standard deck of cards if once 1 card is chosen it is not replaced?

This, of course, is a dependent event since the occurrence of A affects the probability of B. 

Let A = 1st seven chosen

Let B = 2nd seven chosen

P(A and B) = P(A∩B) = P(A) x P(B|A)

The notation with the vertical bar is referred to as a conditional probability. The vertical bar in P(B|A) means "given," so this is read as "the probability that B occurs given that A has occurred."

P(A) = 4/52 (4 sevens in deck of 52 cards)

P(B) = 3/51 (3 remaining sevens in deck with now 51 cards)

P(A∩B) = 4/52 x 3/51 =12/2652 = 1/221

The probability of drawing two sevens consecutively from a standard deck without replacement is 1/221, or approximately 0.45%.

 

Example 2

A bag contains 3 green marbles and 2 red marbles. A marble is drawn, not replaced, and then a second marble is drawn. What is the probability of drawing a green marble followed by drawing a red marble?

Let A = Drawing a green marble on first draw

Let B = Drawing a red marble on second draw

P(A) = 3/5 (3 green marbles; 5 total marbles)

P(B) = 2/4 (2 red marbles; (2 red marbles; 4 total marbles)

P(A) x P(B|A) = 3/5 x 2/4 = 6/20 = 3/10

The probability of drawing a green marble followed by a red marble from a bag containing 3 green and 2 red marbles, without replacement, is 3/10 (or 30%).

B. Conceptual Understanding

Sometimes it can be difficult to figure the correct probability for each event. Using a probability tree diagram can help us to better conceptually understand what is happening. 

Here is the diagram for Example 1

The first red branch represents P(A) of 4/52. Now notice the second red branch which is P(B|A) or stated another way, the probability of B given that A had happened. Since we are stating that A (a seven was drawn) has happened, there are now only 3 sevens left in the deck and only 51 cards remain in total. 

Next is the tree from Example 2

The first red branch represents P(A) which is 3/5. The second branch is P(B|A) stated as the probability of B given that A has occurred. We assume that event A has happened, so there are now only 4 total marbles left to draw but still 2 red marbles. So the P

CK-12: Dependent Events and Sample Spaces

Khan: The general multiplication rule

Mathbits: Independent and Dependent Events

LibreTexts: Probability Using Tree Diagrams and Combinations

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