union of two events.pptx probability of events

probability of events

1. Illustrate the Probability of a Union of Two Events 2.1PROBABILITY OF UNION OF TWO EVENTS.pptx . (2023). SlideShare ; Slideshare . https://www.slideshare.net/slideshow/21probability-of-union-of-two-eventspptx/258738372 ‌

2. Recall Union, Intersection and Complement of a Set Union set – is an element must be in at least one of the sets, could be in both set to combine. “A union B” or “A or B” Ex. AᴗB Universal set = { 1, 2, 3, 4, 5, 6, 7, 8} A={1, 2, 5} B={2, 4, 7} AᴗB={1, 2, 4, 5, 7}

3. Recall Union, Intersection and Complement of a Set Intersection of sets – is an element must be in both sets “A intersects B” or “A and B” Ex. A ᴖ B Universal set = { 1, 2, 3, 4, 5, 6, 7, 8} A={1, 2, 5} B={2, 4, 7} A ᴗ B={2}

4. Recall Union, Intersection and Complement of a Set Complement of a sets – is all element not in the given set “not in set A” Ex. A’ Let A={1, 3, 5, 7, 9} B={1, 4, 6, 7, 9} A’ ᴖ B= {4, 6}

5. COMPLEMENT OF AN EVENT The complement of an event is the set of all outcomes that are NOT in the event. This means that if the probability of an event, A, is P(A), then the probability that the event would not occur is 1-P(A), denoted by P(A’). Thus, P(A’) = 1-P(A)

6. COMPLEMENT OF AN EVENT Example The probability that an event occurs is 2/5. What is the probability that the event does not occur? Solution. P(E’) = 2/5 P(E’) = 1-P(E’) P(E’) = 1-2/5 P(E’) = 5-2/5 P(E’) = 3/5

7. COMPLEMENT OF AN EVENT Example 2. A spinner is divided into six equal sections: green, yellow, blue, violet, red and orange. What is the probability of the spinner not landing on a red? Solution: Sample space (S)= { green, yellow, blue, violet, red and orange} =6 Event (E)= {red} =1 P(E)= n(E)/n(S) =1/6 P(E’)=1-P(E) =1-1/6 =6-1/6 =5/6

8. PROBABILITY OF THE UNION OF: Mutually exclusive events - are those events that do not occur at the same time. In other words, mutually exclusive events are called disjoint events . If two events are considered disjoint events, then the probability of both events occurring at the same time will be zero. P(A and B) = 0 “ The probability of A and B together equals 0 ”impossible”

9. If two events, A and B, are mutually exclusive, then the probability that either A or B occurs is the sum of their probabilities. In symbols, P(A or B) = P(A) + P(B) or P(A  B) = P(A) + P(B)

10. Example 1. What is the probability of a dice showing a 2 or 5?

11. Renz rolled a fair die and wished to find the probability that “the number that turns up is odd or even”.   Given: S =  1,2,3,4,5,6  A =  1,3,5  B =  2,4,6  Formula: P(A  B) = P(A) + P(B) Solution: P(A  B) =  

12. Non-mutually exclusive events – are events that can happen at the same time. In other words, non-mutually exclusive events are called joint events. The probability of the union of two events A and B  written as P(A  B) or P(A or B)  is equal the to sum of the probability of event A  P(A)  and the probability of event B  P(B)  minus the probability of event A and B occurring together  P(A  B)  . In symbols, P(A  B) = P(A) + P(B) – P(A  B)

13. Renz rolled a fair die and wished to find the probability that 2 .“the number that will turn is even or greater than 3”. Given: S =  1,2,3,4,5,6  A =  2,4,5  B =  4,5,6  (A  B) =  4,6  Formula: P(A  B) = P(A) + P(B) – P(A  B) Solution: P(A  B) =  

14. 3. A pair of dice is rolled. What is the probability that the two dice show the same number or that the sum of the numbers is less than 7?

15. A pair of dice is rolled. What is the probability that the two dice show the same number or that the sum of the numbers is less than 7? Given: A =  1-1,2-2,3-3,4-4,5-5,6-6  B = 1-1,1-2,1-3,1-4,1-5,2-1,2-2,2-3,2-4,3-1,3-2,3-3,4-1,4-2 ,5-1  (A  B) =  1-1,2-2,3-3  Formula: P(A  B) = P(A) + P(B) – P(A  B) Solution: P(A  B) =  

16. LET’S SUM IT UP! In solving the probability of the union of two events, if the two sets do not have elements in common, use Mutually exclusive events P(A or B) = P(A) + P(B) o r P(A  B) = P(A) + P(B)   If the two sets have elements in common, use Non-mutually exclusive events P(A or B) = P(A) + P(B) – P(A  B)

17. IT’S YOUR TURN! In one whole sheet of paper, solve the following problem completely. Show complete solution. 5 points each.

18. A bag contains 10 marbles: 4 red, 3 blue, and 3 green. If you randomly select one marble, what is the probability of not selecting a red marble? 2. A die is rolled. What is the probability of not rolling a number greater than 4? 3. In a deck of cards, what is the probability of drawing a heart or a spade?  

19. 4 . A student can either attend a math club meeting or a science club meeting, but not both at the same time. If the probability of attending the math club is 0.3 and the probability of attending the science club is 0.2, what is the probability that the student will attend either the math club or the science club? 5 . In a survey, 40% of students like basketball, and 30% like soccer. If 10% like both sports, what is the probability that a student likes either basketball or soccer?

20. THANK YOU!