Probability of Independent Events 6.4.1.1 Determine the sample space (set of possible outcomes) for a given experiment and determine which members of the sample space are related to certain events. Sample space may be determined by the use of tree diagrams, tables, or pictorial representations.
Independent Events 13.3 Notes Whatever happens in one event has absolutely nothing to do with what will happen next The probability of two independent events, A and B, is equal to the probability of event A times the probability of event B.
I can… Self Assessment Find the probability of 2 independent events 5- I can do it without help & teach others. 4- I can do this with no help, but I don’t know if I can explain it. 3- I can do this with a little help. 2- I can do this with a lot of help! 1- I don’t have a clue.
Independent Events Whatever happens in one event has absolutely nothing to do with what will happen next because: The two events are unrelated OR You repeat an event with an item whose numbers will not change (eg.: spinners or dice) You repeat the same activity, but you REPLACE the item that was removed. The probability of two independent events, A and B, is equal to the probability of event A times the probability of event B.
Independent Events Example: Of the 25 beads in the bag, 15 are yellow, 6 are orange, and 4 are blue. Joe is going to pull one bead out of the bag without looking, replace it, and then pull out another bead. What is the probability of Joe pulling an orange bead from the bag first and then pulling a yellow bead from the bag?
Step 1: Find the probability of each event Independent Events Example: Of the 25 beads in the bag, 15 are yellow, 6 are orange, and 4 are blue. Joe is going to pull one bead out of the bag without looking, replace it, and then pull out another bead. What is the probability of Joe pulling an orange bead from the bag first and then pulling a yellow bead from the bag? Step 1: Find the probability of each event P(orange) = P(yellow) =
Independent Events P(orange, then yellow) = P(orange) • P(yellow) Example: Of the 25 beads in the bag, 15 are yellow, 6 are orange, and 4 are blue. Joe is going to pull one bead out of the bag without looking, replace it, and then pull out another bead. What is the probability of Joe pulling an orange bead from the bag first and then pulling a yellow bead from the bag? Step 2: Multiply the probability of the first event by the probability of the second event. P(orange, then yellow) = P(orange) • P(yellow)
Independent Events Example 2: Suppose you spin each of these two spinners. What is the probability of spinning an even number and a vowel? S T R O P 1 2 3 6 5 4
Step 1: Find the probability of each event Independent Events Example: Suppose you spin each of these two spinners. What is the probability of spinning an even number and a vowel? S T R O P 1 2 3 6 5 4 Step 1: Find the probability of each event P(even) = P(vowel) =
Independent Events P(even, vowel) = P(even) • P(vowel) Example: Suppose you spin each of these two spinners. What is the probability of spinning an even number and a vowel? Step 2: Multiply the probability of the first event by the probability of the second event. P(even, vowel) = P(even) • P(vowel)
Independent Events Example 3: What is the probability of flipping tails then heads? 1st Flip 2nd Flip OUTCOMES H,H H,T P(Outcome) T,H P(T,H) ½ X ½ = ¼ T,T H H H H H T T T T T 1/2 1/2 1/2 1/2 1/2 1/2 1/2
Independent Events Example 4: The spinners are each divided into equal parts. You spin the spinners. Find the probability that the sum is at least 4. 1 2
Independent Events 2 1 Make a Table of Sums 1 2 3 4 5 6 Example 4: The spinners are each divided into equal parts. You spin the spinners. Find the probability that the sum is at least 4. Make a Table of Sums 1 2 3 4 5 6 1 2
Independent Events 2 1 Make a Table of Sums 1 2 3 4 5 6 7 8 Example 4: The spinners are each divided into equal parts. You spin the spinners. Find the probability that the sum is at least 4. Make a Table of Sums 1 2 3 4 5 6 7 8 1 2
Independent Events 2 1 Make a Table of Sums 1 2 3 4 5 6 7 8 Example 4: The spinners are each divided into equal parts. You spin the spinners. Find the probability that the sum is at least 4. Make a Table of Sums 1 2 3 4 5 6 7 8 1 2
Independent Events 2 1 Make a Table of Sums 1 2 3 4 5 6 7 8 Example 4: The spinners are each divided into equal parts. You spin the spinners. Find the probability that the sum is at least 4. Make a Table of Sums 1 2 3 4 5 6 7 8 1 2
Your Turn You and your friend each randomly choose to go swimming or play basketball. What is the probability that both of you choose the same activity? Your Choice Friend’s Choice OUTCOMES P(Outcome) S,S P(S,S) ½ X ½ = ¼ S,B B,S B,B P(B,B) ½ X ½ = ¼ S S S S S B B B B B 1/2 1/2 1/2 1/2 1/2 1/2 1/2 P (total) = ¼ + ¼ = ½
Your Turn 2 4 1 2 = = P(S, S) = P(S) • P(S) = ½ • ½ = ¼ P(B, B) = You and your friend each randomly choose to go swimming or play basketball. What is the probability that both of you choose the same activity? P(S, S) = P(S) • P(S) = ½ • ½ = ¼ P(B, B) = P(B) • P(B) = ½ • ½ = ¼ 2 4 1 2 = =
Your Turn Find the probability P(jack, factor of 12) Slide 19
Your Turn Find the probability P(jack, factor of 12) 1 5 5 8 5 40 1 8 x = 1 8 Slide 20
Your Turn Find the probability P(6, not 5) 1 6 5 6 5 36 x = Slide 21
I can… Self Assessment Find the probability of 2 independent events 5- I can do it without help & teach others. 4- I can do this with no help, but I don’t know if I can explain it. 3- I can do this with a little help. 2- I can do this with a lot of help! 1- I don’t have a clue.
Independent Events 13.3 Notes Whatever happens in one event has absolutely nothing to do with what will happen next The probability of two independent events, A and B, is equal to the probability of event A times the probability of event B.