introduction of probability

Probability

Probability

• Experiment – an activity with observable results.

• Outcomes – the results of an experiment.• Sample Space – the set of all possible

different outcomes of an experiment.

Probability

Ex. a. Tossing a coin once outcome = H S(n)=2

b. tossing three different coins together outcome= HHH S(n)=8

c. rolling a die outcome = 1 S(n) = 6

d. throwing a coin and a die together outcome = H1 S(n)=12

Probability

Event – a subset of a sample space of an experiment.

Subset – Let E be an event of the sample space S. Since every outcome in E is an outcome in S, we say that event E is a subset of S, denoted by

Probability

The Union of two eventsThe union of two events E and F is event

Venn Diagram

Probability

The Union of two eventsExample if set A= { 1,2,3} and set B= { 3,4,5,6}Then AUB=

{1,2,3,4,5,6}

Probability

The Intersection of two events The intersection of two events E and F is the set

Probability

The Intersection of two eventsExample if set A= { 1,2,3} and set B= { 3,4,5,6}Then A =

{3} Example set C = { even integers} and set D={ odd integers}Then C =

Empty set or is called impossible event.

Probability

The complement of an event The complement of an event E is the event

Ec.

Read as “E complement” is the set comprising all the outcomes in the sample space S that are not in E.

Probability

The complement of an event

Probability

The complement of an eventExample If set D = { number of dot/s in a die}

and set E = { 2,4,6 } then Ec is

Ec = { 1,3,5}

Probability

Mutually Exclusive EventsE and G are mutually exclusive event

if

Probability

Mutually Exclusive Events

Probability

Example set E = { even numbers} and set G={ odd numbers}

then E and G are mutually exclusive events.or

C =

Probability1) In rolling a pair of dice (one white and one black), we have the following event. A={(1,6)}B={(1,6),(6,1)}C={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)}D={(1,1),(1,6),(6,1),(6,6)}E={(1,1),(1,3),(5,5)}Question

a) Which events are subsets of the other? A A C, A D, B D, A B D

b) What is AUB? Bc) What is A Ad) What is Be) Which events are mutually exclusive? A and E,B and E

Probability

2) In rolling a pair of dice, determine the events M,N,P, and Q such that the sum of the numbers are 4, 5,6, and 7, respectively.M={(2,2),(1,3),(3,1)}N={(2,3),(3,2),(1,4),(4,1)}P={(3,3),(2,4),(4,2),(1,5),(5,1)}Q={(3,4),(4,3),(2,5),(5,2),(1,6),(6,1)}

Probability – is the measure of how likely an event is to appear. P(E)=

Probability

1. The spinner may stop on any one of the eight numbered sectors of the circles. Use the spinner at the right to find each probability.a. P(2)b. P(white)c. P(9)=d. P(not white)=e. P(white or Red)=

Probability

2. There are 3 red pens, 4 blue pens, 2 black pens, and 5 green pens in a drawer. Suppose you choose a pen at random.a) What is the probability that the pen chosen is

red? 3/14b) What is the probability that the pen chosen is

blue? 2/7c) What is the probability that the pen chosen is

red or black? 5/14

Probability of Independent Events

• The outcome of an independent event is not affected by the outcome of another.

If events A and B are independent, the probability of both events occurring is found by multiplying the probabilities of the events.

P(A and B) = P(A) x P(B) or P(A


Example 1. The spinner at the right is spun twice. Find the probability of each outcome.a) An A and Bb) a B and a C


Example 2. Every end of the month the faculty of Hayna High School goes bowling at the Bowlingan sa Canto. On one shelf of the bowling alley there are 6 green and 4 red bowling balls. One teacher selects a bowling ball. A second teacher then selects a ball from the same shelf. What is the probability that each teacher picked a red bowling ball if replacement is allowed? Solution:P(both red) = P(red) x P(red)

= (4/10)(4/10) = 4/25The probability that both teachers selected a red ball is 4/25.

Probability of Dependent Events

Dependent EventsTwo events are dependent if the outcome of

one of them has an effect on the outcome of another.The Probability of an event B occurring given that an event A has already occurred is P(B/A), and read as, “The probability of B given A”.

P(A and B) = P(A) x P(B/A)

Probability of dependent Events

Example 3.There are 5 blue marbles, 3 red marbles, and 4 black marbles in a box. a) What is P(blue)


Example 3.There are 5 blue marbles, 3 red marbles, and 4 black marbles in a box.

b) What is P(red)



c) What is P(black)



d) What is P(blue and red) if there is no replacement? 5/12 x3/11=5/44



e. ) What is P(blue and black) if there is no replacement? 5/12 x 4/11 =5/33



f) What is P(blue and red) if there is replacement? 5/12 x 3/12 =5/48


• There are eight white socks and five black socks in a drawer.

a. What is the probability that you can pull out a white socks?

b. If you pull one sock out of the drawer and then another, what is the probability that you can pull out 2 white socks?


M.E.E are events in which one or the other of two events, but not both, can appear.

Rule: The Addition RuleIf A and B are M.E.E, then P(A or B)=P(A) + P(B)


Ex. 1.What is the probability of drawing an ace or a king from a standard deck of cards?

Solution: P(ace or King) = P(ace) + P(king)

= =


Example 2. There are 6 girls and 5 boys on the school paper staff. A committee of 5 students is being selected at random to design the editorial illustration of the pilot issue. What is the probability that the committee will have at least 3 boys?Solution:P(at least 3 boys)=P(3 boys)+P(4 boys)+P(5 boys)

= 3b,2g 4b,1g 5b,0g= =

The probability that at 3 boys on the committee is

Probability of Inclusive Events

If two events, A and B, are inclusive, then the probability that either A or B is the sum of their probabilities decreased by the probability of both appearing.P(A or B) = P(A) + P(B)-P(A and B)


Example 1. One die is tossed. What is the probability of tossing a 4 or a number greater than 3?Solution: P(4 or > 3)=P(4)+P(>3)-P(4 and >3)

= P(4 or > 3) =


Example 2. What is the probability of drawing a King or a heart from a deck of cards?Solution:

P(K or <3) = P(K) + P(<3) – P(King of hearts)=

P(K or <3) =

introduction of probability

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