A dresser drawer contains one pair of socks

A dresser drawer contains one pair of socks with each of the following colors: blue, brown, red, white and black. You reach into the sock drawer and choose a pair of socks without looking. The first pair you pull out is red --the wrong color. You replace this pair and choose another pair of socks. What is the probability that you will choose the red pair of socks twice?

You choose the red pair but you replace it and choose another ; choosing a red pair on the first try has no effect on the probability of choosing a red pair on the second try. Therefore, these events are independent.

• Definition: Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring. For ex: • Landing on heads after tossing a coin AND rolling a 5 on a single 6 -sided die. • Choosing a marble from a jar AND landing on heads after tossing a coin. Two events are DEPENDENT when the ocurrance of the second event is affected by the first. Key words are : NO REPLACEMENT • For ex. : choosing a marble, not replacing it back; Then, choosing the second marble. For the second event, you will one less chance.

• To find the probability of two independent events that occur in sequence, find the probability of each event occurring separately, and then multiply the probabilities. Note that multiplication is represented by AND. Multiplication Rule : When two events, A and B, are independent, the probability of both occurring is: P(A and B) = P(A) · P(B)

Probabilities: P(red) = 1/5 P(red and red)= P(red) · P(red) = 1/5 · 1/5 = 1/25

A jar contains 3 red, 5 green, 2 blue and 6 yellow marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. What is the probability of choosing a green and a yellow marble?

Probabilities: P(green) = 5/16 P(yellow) = 6/16 P(green and yellow) = P(green) · P(yellow)= 5/16 · 6/16=30/256 = 15/128

• Multiplication Rule can be extended to work for three or more independent events that occur in sequence. A school survey found that 9 out of 10 students like pizza.

If three students are chosen at random with replacement, what is the probability that all three students like pizza? Probabilities: P(student 1 likes pizza) = 9/10 P(student 2 likes pizza) = 9/10 P(student 3 likes pizza) = 9/10 P(student 1 and student 2 and student 3 like pizza) = 9/10 · 9/10 =729/1000

Summary: The probability of two or more independent events occurring in sequence can be found by computing the probability of each event separately, and then multiplying the results together. Key words: with replacement, put it back

Probability of Dependent Events • If the outcome of the first event affects the outcome of the second event, the events are DEPENDENT. • A bag contains 3 blue and 3 red marbles. Draw a marble, then draw a second marble without replacing the first marble. Find the probability of drawing 2 blue marbles.

Probability of Dependent Events 1. Find P(blue)=3 blue/6 marbles= ½ 2. Find P(blue after blue)= 2 blue/5 marbles=2/5 3. Multiply P(blue, then blue) ½*2/5=1/5