Calculating Probabilities Without a Two-Circle Venn Diagram (part 2)

on Mutually Exclusive Events

Q1. If sample space ={1,2,3,…,9}, event A={2,4,6,8} and Event B={1,3,5} find P(A∪B).
P(A∪B)=P(A)+P(B) … (i).


Put in equation (i), we get

Q2. Two dice are thrown. What is the probability that the sum of the number of dots appearing on them is 4 or 6?
When two dice are rolled then the possible outcomes are


Let A= be the event the sum is 4, then

Let B be the event that the sum is 6, then

Q3. There are 10 girls and 20 boys in a class. Half of the boys and half of the girls have blue eyes. Find the probability that one student chosen as monitor is either a girl or has blue eyes.

Q4. A DVD shop has 180 comedies, 250 drama films, 230 science fiction movies and 120 thrillers. If you select a DVD at random, what is the probability that this movie is a comedy OR a thriller?
No DVD is marked as both a comedy and a thriller, so there is no overlap in events. These are mutually exclusive (but not complementary).
There are 250+230+120=600 DVDs in the sample space.
Use P(A or B)=P(A)+P(B).
P(comedy or thriller)=P(comedy)+P(thriller)

Q5. A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow. The discs are similar in shape and size. A disc is drawn at random from the bag. Calculate the probability that it will be (i) red, (ii) yellow, (iii) blue, (iv) not blue, (v) either red or blue.
There are 9 discs in all so the total number of possible outcomes is 9. Let the events A, B, C be defined as
A: ‘the disc drawn is red’
B: ‘the disc drawn is yellow’
C: ‘the disc drawn is blue’.
(i). The number of red discs =4, i.e., n(A)=4
Hence P(A)=4/9
(ii). The number of yellow discs =2, i.e., n(B)=2
Therefore, P(B)=2/9
(iii). The number of blue discs =3, i.e., n(C)=3
(iv). Clearly the event ‘not blue’ is ‘not C’. We know that
P(not C)=1-⅓=⅔
(v). The event ‘either red or blue’ may be described by the set ‘A or C’. Since, A and C are mutually exclusive events, we have

on Not Mutually Exclusive Events

Let’s read the previous posts

Probability – Mutually Exclusive Events or Not
Probability of Either Event A or B happens, or Both happen

In the previous posts we demonstrated the addition law of probability:
For two events A and B,


which means:

P(either A or B or both) =P(A)+P(B)-P(both A and B)
Q6. The probability of event X is 0.43 and the probability of event Y is 0.24. The probability of both occurring together is 0.10. What is the probability that X or Y will occur?
From the addition rule

P(X or Y)=P(X)+P(Y)-P(X and Y)
Q7. What is the probability of drawing a club or an ace with one single pick from a pack of 52 cards
Step 1: Identify the identity which describes the situation

P(club ∪ ace)=P(club)+P(ace)-P(club ∩ ace)

Step 2: Calculate the answer

Notice how we have used P(C∪A)=P(C)+P(A)-P(C∩A)

Q8. Two dice are rolled; find the probability of getting doubles or a sum of 6.
Let A= getting doubles; then P(A)=6/36 since there are six ways to get doubles and let B = getting a sum of 6. Then P(A)=5/36 since there are five waysto get a sum of 6→ (5, 1), (4,2), (3,3), (2,4), and (1,5). Let A= and B=thenumber of ways to get a double and a sum of 6. There is only one way forthis event to occur→namely (3,3); then P(A and B)=1/36. Hence,

From 100 cards, numbered from 1 to 100, one is selected at random. Find the probability that the card selected is even or less than 20.
Some cards are both even and less than 20 (i.e. 2, 4, 6, 8, 10, 12, 14, 16, 18).

P(even or <20)=P(even)+P (<20)-P(even and <20)

Q10. A card is drawn from a deck of 52 playing cards what is the probability that it is a diamond card or an ace:
Here n(S)=52

Q11. From the employees of a company, 5 persons are selected to represent them in the managing committee of the company. Particulars of five persons are as follows:

A person is selected at random from this group to act as a spokesperson. What is the probability that the spokesperson will be either male or over 35 years?
Let E be the event in which the spokesperson will be a male and F be the event in which the spokesperson will be over 35 years of age.
Accordingly, P(E)=⅗ and P(F)=⅖
Since there is only one male who is over 35 years of age,


We know that

P(E∪F) =P(E)+P(F)-P(E∩F)

Thus, the probability that the spokesperson will either be a male or over 35 years of age is ⅘.
Let’s red the post Calculating Probability Without Two-Circle Venn Diagram (part 1)


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