Probability of an event
• The probability of an event A is the sum of the probabilities of the individual outcomes of which it is composed. It is denoted P(A).
• If an event A is A={e1 ,e2 ,…,ek },
then • P(A)=P(e1 )+P(e2 )+ ⋅ ⋅ ⋅ +P(ek)
Example:
Q) A die is called “balanced” or “fair” if each side is equally likely to land on top. Assign a probability to each outcome in the sample space for the experiment that consists of tossing a single fair die. Find the probabilities of the events E: “an even number is rolled” and T: “a number greater than two is rolled.”
--> • Solution:
• S={1,2,3,4,5,6}.
• Probability of each possible outcome is 1/6.
• Since E={2,4,6}, P(E)=1⁄6+1⁄6+1⁄6=3⁄6=1⁄2.
• Since T={3,4,5,6}, P(T)=4⁄6=2⁄3
Q) A box contains 10 white and 10 black marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, two marbles in succession and noting the color each time.
-->•Solution:
S={bb,bw,wb,ww}
• In the given experiment, list the outcomes that comprise each of the following events.
1. At least one marble of each color is drawn.
2. No white marble is drawn
-->1. {bw,wb} 2. {bb}
Q)The following two-way contingency table gives the breakdown of the population in a particular locale according to age and tobacco usage: •A person is selected at random. Find the probability of each of the following events. (1) The person is a smoker. (2) The person is under 30. (3) The person is a smoker who is under 30
--> •Solution:
1. 0.05 + 0.20 = 0.25
2. 0.05 + 0.20 = 0.25
3. 0.05
Q)Find the probability of getting a numbered card when a card is drawn from the pack of 52 cards
-->•Solution: 1. Total Cards = 52.
2. Numbered Cards = (2, 3, 4, 5, 6, 7, 8, 9, 10) 9 from each suit 4 × 9 = 36
3. P (E) = 36/52 = 9/13
Q) There are 5 green 7 red balls. Two balls are selected one by one without replacement. Find the probability that first is green and second is red
•Solution:
P (G) × P (R) = (5/12) x (7/11) = 35/132
Q)A sample space is S={a,b,c,d,e}. Identify two events as U={a,b,d} and V={b,c,d}. •Suppose P(a) and P(b) are each 0.2 and P(c) and P(d) are each 0.1.
•Determine what P(e).
•Find P(U).
•Find P(V).
--> •Solution: 1. 0.4
2. 0.5
3. 0.4
Complement of an event
• The complement of an event A in a sample space S, denoted Ac, is the collection of all outcomes in S that are not elements of the set A.
• It corresponds to negating any description in words of the event A.
• Probability Rule for Complements • P(Ac)=1−P(A)
Q) Two events connected with the experiment of rolling a single die are E: “the number rolled is even” and T: “the number rolled is greater than two.” Find the complement of each.
-->• Solution:
• In words the complements are described by “the number rolled is not even” and “the number rolled is not greater than two.”
• Of course easier descriptions would be “the number rolled is odd” and “the number rolled is less than three.”
• In the sample space S={1,2,3,4,5,6} the corresponding sets of outcomes are E={2,4,6} and T={3,4,5,6}. The complements are Ec={1,3,5} and Tc={1,2}