Which formula is right to use, P(A or B) = P(A) + P(B) - P(A∩B) or P(A ∪ B) = P(A) + P(B) - P(A∩B)?

[arzourwayne]

Hi there Math Expert,

I really need your help, I'm stuck with this question and indeed, I am new to this subject. With that, I appreciate for someone who willing to help.

Obama is a member of the musical club in local university. The probability that Obama plays a piano is 1/4 and the probability that he plays a bass drum is 5/8. If the probability that he plays both of these instruments is 5/24, what is the probability that he plays the piano or that he plays the bass drum?

So, this question and subject is new to me. I don't really know which formula should use, either addition rules (as shown on title) or maybe other rules. Hopefully, someone out there willing to help me with this. Thanks in advance.

 

[blamocur]

The formula -- there are actually two identical ones -- in the title looks fine to me. Here is an intuitive explanation: when you compute [imath]P(A)+P(B)[/imath] you almost get [imath]P(A\cup B)[/imath], except that you count [imath]P(A \cap B)[/imath] twice: once in [imath]P(A)[/imath] and once in [imath]P(B)[/imath]; to get it right you subtract one of them.

 

[arzourwayne]

The formula -- there are actually two identical ones -- in the title looks fine to me. Here is an intuitive explanation: when you compute [imath]P(A)+P(B)[/imath] you almost get [imath]P(A\cup B)[/imath], except that you count [imath]P(A \cap B)[/imath] twice: once in [imath]P(A)[/imath] and once in [imath]P(B)[/imath]; to get it right you subtract one of them.

So that is to say, 1/4 + 5/8 - (1/4 ∩ 5/8), is that right?

 

[JeffM]

These symbols, [imath]\cup[/imath] and [imath]\lor[/imath], mean “or.”

These symbols, [imath]\cap[/imath] and [imath]\land[/imath], mean “and.”

[math]0 \le P(X) \le 1.\\ P(X) = 0 \iff X \text { is impossible.}\\ P(X) = 1 \iff X \text { is certain.}[/math]
[math]P(A \cup B) = P(A) + P(B) - P(A \cap B).[/math]
[math]P(M \ | \ N) \text { means the probability that M happens if N happens and } P(N) > 0.[/math]
[math]P(A \ | \ B) * P(B) = P(A \cap B) = P(B \ | \ A) * P(A).[/math]
[math]P(A \cap B) = 0 \iff A \text { and } B \text { are mutually exclusive.}\\ P(A \cup B) = 1 \iff A \text { and } B \text { are exhaustive.}[/math]
Just about all you need to know about probabilities is summarized above.

 

[arzourwayne]

T

These symbols, [imath]\cup[/imath] and [imath]\lor[/imath], mean “or.”

These symbols, [imath]\cap[/imath] and [imath]\land[/imath], mean “and.”

[math]0 \le P(X) \le 1.\\ P(X) = 0 \iff X \text { is impossible.}\\ P(X) = 1 \iff X \text { is certain.}[/math]
[math]P(A \cup B) = P(A) + P(B) - P(A \cap B).[/math]
[math]P(M \ | \ N) \text { means the probability that M happens if N happens and } P(N) > 0.[/math]
[math]P(A \ | \ B) * P(B) = P(A \cap B) = P(B \ | \ A) * P(A).[/math]
[math]P(A \cap B) = 0 \iff A \text { and } B \text { are mutually exclusive.}\\ P(A \cup B) = 1 \iff A \text { and } B \text { are exhaustive.}[/math]
Just about all you need to know about probabilities is summarized above.

Thank you for your effort sir, would you mind to solve my question, so I can understand how it works. I literally have no idea what is that thing.

 

[JeffM]

What is WHAT THING?

Let A = the case such that he plays the piano.

According to the statement of the problem, what is P(A)?

Let B = the case that je plays the bass drum.

According to the statement of the problem, what is P(B)?

According to the statement of the problem, what is the probability of (A and B)?

Now using the rules given in post # 4, what is the probability of (A or B)

 

[blamocur]

T

Thank you for your effort sir, would you mind to solve my question, so I can understand how it works. I literally have no idea what is that thing.

Hint: what is the value of [imath]P(A\cap B)[/imath] ?

 

[pka]

I really need your help, I'm stuck with this question and indeed, I am new to this subject. With that, I appreciate for someone who willing to help.
Obama is a member of the musical club in local university. The probability that Obama plays a piano is 1/4 and the probability that he plays a bass drum is 5/8. If the probability that he plays both of these instruments is 5/24, what is the probability that he plays the piano or that he plays the bass drum?

First I want to say that you are mixing two very different notations. One belongs to a logic course while the other is used in a course on probability .
The symbol [imath]\vee[/imath] stands for or as in [imath](p\vee q)[/imath] "p or q" which has only two values: true or false.
The statemet [imath](p\vee q)[/imath] is true if and only if at least one of [imath](p\text{or } q)[/imath] is true.
Now consider what a probability space is about. The space is composed of a set of events{ subsets of the universe} and a measure on that sets of events [imath]\mathcal{P}(A)[/imath] where [imath]0\le\mathcal{P}(A)\le 1[/imath].
So lets say that in the given question the space of events are about belonging to a music club.
If [imath]A[/imath] is the event of playing a piano and [imath]B[/imath] is the event of playing a bass drum.
We are given that [imath]\mathcal{P}(A)=\dfrac{1}{4}[/imath], [imath]\mathcal{P}(B)=\dfrac{5}{8}[/imath] and [imath]\mathcal{P}(A\cap B)=\dfrac{5}{24}[/imath].
In your probability class you have learned that [imath]\mathcal{P}(A\cup B)=\mathcal{P}(A)+\mathcal{P}( B)-\mathcal{P}(A\cap B)[/imath]
Can you finish?
[imath][/imath][imath][/imath]
[imath][/imath]
[imath][/imath]
[imath][/imath]

 

[Steven G]

what is the probability that he plays the piano or that he plays the bass drum?

Let A represent playing the piano and B represent playing the bass. So you want to find p(A or B). I would use the formula that starts with p(A or B) = .....

Isn't that obvious? Deciding on which equation to use has nothing to do with probability.

 

[pka]

what is the probability that he plays the piano or that he plays the bass drum?
Let A represent playing the piano and B represent playing the bass. So you want to find p(A or B). I would use the formula that starts with p(A or B) = .....
Isn't that obvious? Deciding on which equation to use has nothing to do with probability.

It seems to me that it is obvious the whole post has everything to do with probability.