Upcoming SlideShare
×

# P2 Probability

3,284 views

Published on

Published in: Technology, Education
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

### P2 Probability

1. 1. ppr maths nbk PROBABILITY II Notes: Probability of an event is the likelihood an event to occur. 1. The probability that event A occurs = Number of outcomes of A Total number of possible outcomes P (A) = n(A) n(S) and the value of probality A is between 0 and 1 = 0 ≤ P(A) ≤ 1 2. S - is the sample space ( the set of possible outcomes) 3. (a) If A is an impossible event , then P(A) = 0 (b) If A is a confirmed event , then P(A) = 1 4. example 5 7 8 10 13 16 Probability of an odd number is chosen S = { 5, 7 ,8, 10, 13, 16 } Let A be the event of getting an odd number n (S ) 3 = 6 1 = 2 1
2. 2. ppr maths nbk 5. If A is an event , the A’ is the complementary event of A , that is P ( A’) = 1 − P(A) example, A box contains a total of 100 red and green marbles. The probability of 3 choosing a red marble is , find the probability of choosing a green marble. 5 Let A be the event of choosing a red marble and A’ be the event of choosing a green marble. P( A’) = 1 − P(A) 3 = 1 − 5 2 = 5 2 Therefore , the probability of getting a green marble is 5 6. Probability of a Combined Event. A combined event is made up of two or more events that happen in either an “or” or and “and” condition. Outcomes of a Combined Events, ( 1) Event A or Event B = A U B (2) Event A and Event B = A I B Example , Two dice are rolled at the same time . Let A = Event of obtaining two even numbers in the two dice B = Event that the sum of the numbers from the dice is less than 10 2
3. 3. ppr maths nbk Solution All possible outcomes when two dice are rolled S ={(1,1), (2,1), (3,1), (4,1),(5,1),(6,1), (1,2), (2,2), (3,2), (4,2), (5,2),(6,2), (1,3), (2,3), (3,3), (4,3), (5,3), (6,3), (1,4), (2,4), (3,4), (4,4), (5,4), (6,4) (1,5), (2,5), (3,5), (4,5), (5,5), (6,5),(1,6), (2,6), (3,6), (4,6), (5,6), (6,6) } n(S) = 36 A = Event of obtaining two even numbers in the two dice A = { (2,2), (2,4), (2,6),(4,2), (4,4), (4,6), (6,2), (6,4), (6,6) } n(A) = 9 B = Event that the sum of the numbers from the two dice is less than 10 B = { (1,1), (1,2), 1,3), (1,4),(1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6) (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1),(4,2),(4,3), (4,4), (4,5), (5,1), (5,2), (5,3), (5,4), (6,1), (6,2), (6,3) } n(B) = 30 The outcomes of combined events (a) A or B = A U B = {(1,1), (1,2), (1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6), (3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6), (5,1),(5,2),(5,3),(5,4),(6,1),(6,2),(6,3),(6,4),(6,6) } n ( A U B ) = 33 P ( AU B ) = n ( AU B ) n(S) 33 11 = = 36 12 3
4. 4. ppr maths nbk (b) A and B = A I B = {(2,2), (2,4), (2,6), (4,2),(4,4),(6,2) n( A I B)= 6 n( A I B) = n( A I B) n (S) 6 = 36 1 = 6 7. If A I B ≠ Ø , therefore P(A U B ) = P(A) + P(B) − P( A I B ) If A I B = Ø , therefore P(A U B ) = P(A) + P(B) example If a dice is rolled, the possible outcomes will be S = {1, 2, 3, 4, 5, 6 } and find the probability that ( i). number 4 or odd number is obtained (ii). number 2 or even number is obtained (iii). even number or prime number is obtained ( i). Let A = event of obtaining number 4. Let B = event of obtaining odd number A= {4} B = { 1, 3, 5 } n( A) = 1 n( B ) = 3 P(A U B ) = P(A) + P(B) = n( A) + n( B ) n(S) n(S) 1 3 = + 6 6 4 = 6 2 = 3 4
5. 5. ppr maths nbk (ii). Let A = event of obtaining number 2 . Let B = event of obtaining even number A={2} B = { 2, 4, 6 } n( A ) = 1 n(B) = 3 P(A U B ) = P(A) + P(B) − P( A I B ) = n( A) + n( B ) − P ( A I B ) n(S) n(S) n(S) 1 3 1 = + − 6 6 6 3 = 6 1 = 2 (iii). Let A = event of obtaining even number. Let B = event of obtaining prime = { 2, 4, 6 } number n(A) = 3 = { 2, 3, 5 } (B) = 3 P(A U B ) = P(A) + P(B) − P( A I B ) = n( A) + n( B ) − P ( A I B ) n(S) n(S) n(S) 3 3 1 = + − 6 6 6 5 = 6 5
6. 6. ppr maths nbk Exercise 1 Paper 2 1. A box contains a total of 42 yellow and green marbles. 14 of them are yellow. A marble is picked randomly from the box. Find the probability of picking a green marble 2. A tray contains some soft-boiled eggs and some hard-boiled eggs. The probability 2 of choosing a soft –boiled egg is . Find the probability of choosing a hard-boiled 5 egg. 3. There are 5 English books, 4 Mathematics books and 3 Science books on a table. A book is choosen at random from the table, find the probability of choosing a Mathematics or Science book. 3 4. The probability of winning a chess competition between team P and team Q are 7 7 and respectively . Find the probability that 15 (a) team P loses , (b) team P wins but team Q loses, (c) both teams win, (d) at least one team wins. 5. Bag A contains six red balls and two purple balls. Bag B contains eight red balls and four purple balls. A ball is taken out randomly from bag A followed by another ball from bag B . Write the probability in fraction form of (a) both balls being red , (b) both balls being purple , (c) both balls being of the same colour , (d) both balls being different colours. 6. A parcel consists of five cards numbered 1 to 5. A card is taken out randomly and its number is recorded. After replacement, another card is taken out from the parcel and its colour is recorded. Find the probability that ( a ) the sum of the numbers of the two cards selected is an even number. ( b ) the first card is ‘ 1 ‘ and the second card is an odd number, ( c ) the number of the first card divided by the number of the second card is less than 1 6