Computing Probabilities By Using Equally Likely Outcomes - Finite Math
2.4Computing Probabilities By Using Equally Likely Outcomes
Calculating Probabilities This section combines what you learned from 2.1-2.3 In 2.1, you learned that if there are n elements equally likely in the sample space, then each element has probability 1/n. In 2.2 and 2.3, you learned how to determine the number of elements in counting arrangements of permutations and combinations Therefore, the probability of an event is the number of elements in the event, divided by the number of elements in the sample space, given each element is equally likely (which will be in problems you encounter in this class)
Let’s do an example. I have 7 coins (2 dimes and 5 quarters). I select 2 at random. What’s the probability that they are both quarters? We need two pieces of information here: # of elements in the event of drawing 2 quarters, and the # of elements in the sample space of draw any two coins. This is a combination problem. # elements (2Q) = C(5,2) = 10 (2Q out of 5) # elements (any 2 coins) = C(7,2) = 21 (2 coins out of 7) Therefore: Pr(2Q) = [# element (2Q)]/[# element SS] = 10/21
Quiz 2.4 #1 An urn has 7 balls – 4 red and 3 green. Jimmy picks 3 balls from the urn. What is the probability that all 3 are red? A. 1/35 B. 4/35 C. 10/35
Quiz 2.4 #1 An urn has 7 balls – 4 red and 3 green. Jimmy picks 3 balls from the urn. What is the probability that all 3 are red? A. 1/35 B. 4/35 C. 10/35 Answer: B
More advanced problem Remember from 2.3 you learned about calculating elements with multiple pools and multiple scenarios. Let’s put that to use with calculating probabilities. Harry and Sally are auditioning for a play along with 4 other males and 2 other females. The play requires 3 distinct male roles and 2 distinct female roles. If the actors are selected at random, what is the probability both Harry and Sally are chosen?
This is a permutation problem with multiple pools. I will be using the slot method to solve this: To determine the probability of an event, again we’ll use the formula: # of elements in event (Harry is in, Sally is in) H(1) x 4 x 3 x S(1) x 2 = 24 Since Harry can also be M2 or M3, we multiply it by 3. Also, Sally can be F2 so we multiply by 2 as well. So the total # of elements in event is 24 x 3 x 2 = 144
# of elements in sample space (any 3 males, and 2 females) 5 x 4 x 3 x 3 x 2 = 360 The answer: 144/360
Quiz 2.4 #2 You and two other friends are in the “I love finite” club with 10 members. The 3 of you are running for the club committee, which consists of 4 members. If the members are chosen at random, what is the probability that at least 2 of you will be elected? A. 21/210 B. 42/210 C. 70/210
Quiz 2.4 #2 You and two other friends are in the “I love finite” club with 10 members. The 3 of you are running for the club committee, which consists of 4 members. If the members are chosen at random, what is the probability that at least 2 of you will be elected? A. 21/210 B. 42/210 C. 70/210 Answer: C
Summary How to find the probability of an event
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