MHB Probability of AND vs. OR: Understanding the difference and formulas

[mathdad]

Can someone explain in simple terms the difference between the probability of AND and the probability of OR.
Can you provide an example for each? Can you please explain the AND/OR formulas for each probability found in most textbooks?

 

[Jameson]

Can someone explain in simple terms the difference between the probability of AND and the probability of OR.
Can you provide an example for each? Can you please explain the AND/OR formulas for each probability found in most textbooks?

I can try to explain this conceptually.

In basic probability, AND is more restrictive than OR. If we have two events A,B then AND requires both events to occur while OR just requires one of them to occur.

When mathematically using these terms, for OR we usually end up adding two probabilities together, increasing the total probability. When using AND we usually multiply two probabilities, which results in something smaller.

An example could be: A is the event that tomorrow is rainy, B is the event that I hear my favorite song on the radio. Assuming they have nothing to with each other, then A AND B occurring means tomorrow is rainy and I hear my favorite song. Both must occur or this AND isn't true. A OR B occurring means tomorrow is rainy, tomorrow I hear my favorite song, or both happen. Three possibilities and more likely than both happening alone.

 

[mathdad]

I can try to explain this conceptually.

In basic probability, AND is more restrictive than OR. If we have two events A,B then AND requires both events to occur while OR just requires one of them to occur.

When mathematically using these terms, for OR we usually end up adding two probabilities together, increasing the total probability. When using AND we usually multiply two probabilities, which results in something smaller.

An example could be: A is the event that tomorrow is rainy, B is the event that I hear my favorite song on the radio. Assuming they have nothing to with each other, then A AND B occurring means tomorrow is rainy and I hear my favorite song. Both must occur or this AND isn't true. A OR B occurring means tomorrow is rainy, tomorrow I hear my favorite song, or both happen. Three possibilities and more likely than both happening alone.

Great reply. Can you provide an actual AND/OR problem for each?

 

[HallsofIvy]

He just did: "A is the event that tomorrow is rainy, B is the event that I hear my favorite song on the radio. Assuming they have nothing to with each other, then A AND B occurring means tomorrow is rainy and I hear my favorite song. Both must occur or this AND isn't true. A OR B occurring means tomorrow is rainy, tomorrow I hear my favorite song, or both happen. Three possibilities and more likely than both happening alone."

If you want to make this a probability problem, assume that P(A)= 0.2 (there is a 20% chance that it will rain tomorrow) and P(B)= 0.05 (there is a 5% chance he will hear his favorite song). Then (assuming these are independent events) the probability he will hear his favorite song and it will rain tomorrow is P(A)*P(B)= (0.2)(0.05)= 0.010 or 1%. The probability he will either hear his favorite song or it will rain tomorrow is P(A)+ P(B)- P(A)*P(B)= 0.2+ 0.05- (0.2)(0.05)= 0.25- 0.01= 0.24.

Another way to do this: Imagine 100 such days. In 20% of them, 20, it rains. In 5% of them, 5, he hears his favorite song. One of those days, both happen. So we have 19 days on which it rains but he does not hear his favorite song, 4 days on which he hears his favorite song but it does not rain, and 1 day on which he hears his favorite song and it rains. That is a total of 19+ 4+ 1= 20+ 5- 1= 24 days on which he either hears his favorite song or it rains. That is a probability of 24/100= 0.24 again.