If a and b are two events, then what is the probability of occurrence of either event a or event b?

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Suppose A and B are two independent events, associated with a random experiment. The probability of occurrence of event A or B is 0.8, while the probability of occurrence of event A is 0.5.Determine the occurrence of the probability of Event B.

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Probability Rules
1. What is the Probability of A and B?
2. What is the Probability of A or B?

I have searched a lot for this question. I am new to probability. How can we convert the basic formulae for this?

P(A)+P(B)-P(A intersection B)= P(A union B)

Thanks for the help in advance. Please down mark this question as after this my account will get blocked

Another word for probability is a possibility. Probability is a branch of mathematics, concerning how likely an event is to occur. The probability of number is indicated from zero to one. In mathematics, Probability has been described to predict how likely there are chances of occurring a likely event. The meaning of probability is the chance to which something is probably or certainly happen. Given below are the terminologies used in probability,

Set Language	Set Notation
Subset A (or event A)	eg, A
Complement of A	Ac
Union of A and B	A ∪ B
Intersection of A and B	A ∩ B or AB
A and B are disjoint (mutually exclusive)	P(A ∩ B) = 0.
A is a subset of B	A ⊆ B

Probability Rules

There are different probability rules like a complementary rule, difference rule, inclusion-exclusion rule, conditional probability, etc. Let’s take a look at these rules in detail,

Complement Rule: According to this rule, the possibility that A does not occur is equal to the possibility that the complement of event A occurs.

Formula ⇒ P(Ac) = 1 – P(A).

Difference Rule: According to this formula if A is a subset of B, then the possibility of B occurring but not A is,

⇒ P(B) – P(A) = P(B A^c).

Inclusion-Exclusion Rule: According to this rule, the possibility of either A or B (or both) occurring is,

⇒ P(A U B) = P(A) + P(B) – P(AB).

Conditional Probability: According to this probability the measure of the probability of an event occurring given that another event has already occurred = P(A|B). In other words, among those instance where B has occurred, P(A|B) is the proportion of cases in which event A occurs.
Multiplication Rule: According to this rule, the possibility of both A and B occurring is equal to the possibility that B occurs times the conditional possibility that A occurs given that B occurs.

⇒ P(AB) = P(B) P(A|B).

Consequently, the conditional probability is given by P(A|B) = P(AB)/P(B). Similarly, the possibility that A occurs times the conditional possibility that B occurs given that A has: P(A) P(B|A) = P(AB), so, P(B|A) = P(AB)/P(A).

Bayes’ Rule: This formula relates the depending probability of B given A to the depending probability of A given B.

⇒ P(B|A) = P(B) P(A|B) / P(A)

Average Formula: Say that the set A can be completely separated into n combined exclusive subsets. Then the all-inclusive probability of A is equal to the average probability of A in the subgroup weighted by the probability of those subgroup:

⇒ P(A) = P(A|B1) P(B1) + P(A|B2) P(B2) + … + P(A|Bn) P(Bn)

Independence: According to this rule if the possibility of A does not depend on whether or not B occurs, then we say that A and B are independent which means they are not dependent on each other.

For independent events ONLY,

⇒ P(A|B) = P(A)

⇒ P(AB) = P(A) P(B)

Solution:

Let’s consider A and B are the likely happening event. According to Inclusion-Exclusion Rule:
The probability of either A or B (or both) occurring is,
⇒ P(A U B) = P(A) + P(B) – P(AB).
For example: If a coin is tossed two times what is the probability of getting either head or tail or both tails.
When a coin is tossed, either a HEAD or a TAIL is obtained.
The Probability of either is the same, which is 0.5 or 1⁄2.
When two coins flipped the possible outcome are:
Note that there is an equal possibility of happening any of the four combinations as
P(HEADS) = P(TAILS) = 0.5
There are 4 possible outcomes i.e., (H, H), (H, T), (T, H), and (T, T). Only one option is of the four is (TAILS, TAILS), so P(FLIP 1 and FLIP 2 = TAILS, TAILS).
= (1/4) = 0.25 = 25%
Now apply the formula: The probability of either A or B (or both)events occurring is
⇒ P(A U B) = P(A) + P(B) – P(AB).
= 1⁄2 + 1⁄2 – 1⁄4
= 2⁄2 – 1⁄4
= 0.75

Question1: If the probability of having green eyes is 10%, the probability of having brown hair is 75%, and the probability of being a green-eyed brown-haired person is 9%, let us assume, A as green eyes and B as brown hair, what is the probability of:

Not having green eyes?
Have green eyes but not brown hair?
Have green eyes and/or brown hair?

Solution:

Formula: P(Ac)
= 1 – P(A) (According to Complement Rule)
= 1 – 10%

= 0.9 or 90%
Having green eyes but not brown hair?
Formula: P(A) – P(AB)
= 10% – 9%
=0.091 or 9.1%
Having green eyes and/or brown hair?
Formula: P(A U B)
= P(A) + P(B) – P(AB) (According to inclusion-exclusion rule)
= 10% + 75% – 9%
= 0.15925 or 15.925%

Question 2:If the probability of having black shoes is 9%, the probability of having a brown shirt is 75%, and the probability of a person wearing a black-shoes brown shirt is 10%, let us consider, A as black shoes and B as a brown shirt, what is the probability of:

Not having black shoes?
Have black shoes but not a brown shirt?
Have black shoes and/or a brown shirt?

Solution:

Formula: P(Ac)
= 1 – P(A) (According to Complement Rule)
= 1- 9%
= 0.91 or 91%
Having black shoes but not brown shirt?
Formula: P(A) – P(AB)
= 9% – 10%
= 0.081 or 8.1%
Having black shoes and/or brown shirt?
Formula: P(A U B)
= P(A) + P(B) – P(AB) (According to inclusion-exclusion rule)
= 9% + 75% – 10%
= 0.14175 or 14.175%

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Watch the video for a few quick examples of how to find the Probability of A and B / A or B:

Probability of A or B (also A and B)

Watch this video on YouTube.

Can’t see the video? Click here.

You may want to read this article first: Dependent or Independent Event? How to Tell the Difference.

Probability of A and B.
Probability of A or B.

A Venn diagram intersection shows events a and b happening together.

1. What is the Probability of A and B?

The probability of A and B means that we want to know the probability of two events happening at the same time. There’s a couple of different formulas, depending on if you have dependent events or independent events.

Formula for the probability of A and B (independent events): p(A and B) = p(A) * p(B).

If the probability of one event doesn’t affect the other, you have an independent event. All you do is multiply the probability of one by the probability of another.

Examples

Example 1: The odds of you getting promoted this year are 1/4. The odds of you being audited by the IRS are about 1 in 118. What are the odds that you get promoted and you get audited by the IRS?

Solution:
Step 1: Multiply the two probabilities together: p(A and B) = p(A) * p(B) = 1/4 * 1/118 = 0.002.

That’s it!

Example 2: The odds of it raining today is 40%; the odds of you getting a hole in one in golf are 0.08%. What are your odds of it raining and you getting a hole in one?

Solution:
Step 1: Multiply the probability of A by the probability of B. p(A and B) = p(A) * p(B) = 0.4 * 0.0008 = 0.00032.

That’s it!

Formula for the probability of A and B (dependent events): p(A and B) = p(A) * p(B|A)

The formula is a little more complicated if your events are dependent, that is if the probability of one event effects another. In order to figure these probabilities out, you must find p(B|A), which is the conditional probability for the event.

Example question: You have 52 candidates for a committee. Four are persons aged 18 to 21. If you randomly select one person, and then (without replacing the first person’s name), randomly select a second person, what is the probability both people will be between 18 and 21 years old?

Solution:
Step 1: Figure out the probability of choosing an 18 to 21 year old on the first draw. As there are 52 possibilities, and 4 are aged 18 to 21, you have a 4/52 = 1/13 chance.

Step 2: Figure out p(B|A), which is the probability of the next event (choosing a second person aged 18 to 21) given that the first event in Step 1 has already happened.
There are 51 people left, and only 3 are aged 18 to 21 now, so the probability of choosing a young adult again is 3/51 = 1 / 17.

Step 3: Multiply your probabilities from Step 1(p(A)) and Step 2(p(B|A)) together:
p(A) * p(B|A) = 1/13 * 1/17 = 1/221.

Your odds of choosing two people aged 18 to 21 are 1 out of 221.

2. What is the Probability of A or B?

The probability of A or B depends on if you have mutually exclusive events (ones that cannot happen at the same time) or not.

If two events A and B are mutually exclusive, the events are called disjoint events. The probability of two disjoint events A or B happening is:

p(A or B) = p(A) + p(B).

Example question: What is the probability of choosing one card from a standard deck and getting either a Queen of Hearts or Ace of Hearts? Since you can’t get both cards with one draw, add the probabilities:
P(Queen of Hearts or Ace of Hearts) = p(Queen of Hearts) + p(Ace of Hearts) = 1/52 + 1/52 = 2/52.

If the events A and B are not mutually exclusive, the probability is:

(A or B) = p(A) + p(B) – p(A and B).

Example question: What is the probability that a card chosen from a standard deck will be a Jack or a heart?
Solution:

p(Jack) = 4/52
p(Heart) = 13/52
p(Jack of Hearts) = 1/52

So:
p(Jack or Heart) = p(Jack) + p(Heart) – p(Jack of Hearts) = 4/52 + 13/52 – 1/52 = 16/52.

References

Salkind, N. (2019). Statistics for People Who (Think They) Hate Statistics 7th Edition. SAGE.

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