Probability of Multiple Events

 Today’s standard: CCSS.MATH.CONTENT.7.PS.8.A Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs CCSS.MATH.CONTENT.7.PS.8.A  Students should answer at least 75% of questions correctly on an exit ticket.

 Independent events do not affect each other’s probabilities at all.  Example: Whether a coin lands on heads or tails does not affect the number that a die lands on.

 Non-example: Picking two cards out of a deck. Since the first card cannot be picked again, the probabilities change the second time.  There is a 13/52 chance of drawing a Spade card from a deck. But then on the second draw, the probability of Spades will be 12/51 if the first card was Spades or 13/51 if it wasn’t. Therefore, these are dependent events.

 Are the following independent or dependent events?  P(rain in Seattle) and P(Bob late in LA)  Picking a marble from a box, putting it back, and picking again.  Picking a marble from a box, setting it aside, and picking again. Independent because whether it rains or not in Seattle does not affect if Bob will be late in LA or not. Independent because when you put the marble back it is like it started so the second pick is not affected by the first. Dependent because the first marble that you remove affects the amounts of the marbles still in the box.

 What is the probability that two coins both land on heads?  To help answer we will use a tree diagram, which we will see more of tomorrow.

This shows all the possible outcomes when two coins are flipped.

 We need the first coin to be heads, so only keep the outcomes where Coin 1 is Heads (½ of them) and get rid of the other half.

 Of the remaining half, only half of those have Coin 2 Heads.  What is half of a half? ½ × ½ = ¼  There is a ¼ chance that two coins will both be heads.

 To find the probability that multiple independent events will happen, multiply their individual probabilities together.

 Example: I roll a die twice. What is the probability I get 4, then a 3?  P(Roll a 4) = P(Roll a 3) =  P(Roll a 4, then a 3) =

 Example: There are 5 red and 3 blue wedges on a spinner. What is the probability I get blue, then red?  P(Blue) = P(Red) =  P(Blue, then Red) =

 Example: There is a 20% chance of rain in LA, 40% chance in NY, and 30% chance in London. What is the probability it rains in all three cities?  0.2 × 0.4 × × = 2.4%

 To find the probability that multiple dependent events will happen, again multiply the individual probabilities. Be careful to adjust the probabilities based on the previous results.

 Example: There are 4 green and 6 orange marbles. I pick two of them. What is the probability they are both green?  First pick: P(Green) =  Now there are 3 green and 6 orange marbles left.  Second pick: P(Green) =  P(Both Picks Green) =