# Easy probability (venn diagrams/conditional pr)

• t_n_p
In summary, the probabilities for the Venn diagram are as follows: Pr(A∩B)=1/11, Pr(AUB)=25/33, and Pr(AUB')=6/11. For the card problem, the probability of choosing the ace of spades given at least one ace is chosen can be found using the conditional probability formula, Pr(A|B) = Pr(A∩B)/Pr(B), where Pr(A) is the probability of choosing the ace of spades and Pr(B) is the probability of choosing at least one ace. To find Pr(B), we can use the complement rule and subtract the probability of choosing no aces from 1.
t_n_p
Q1. Venn diagrams.

[PLAIN]http://img826.imageshack.us/img826/3194/23872542.jpg
Find:
a) Pr(A∩B)
b) Pr(AUB)
c) Pr(AUB')

a) this is easy, Pr(A∩B)=3/33=1/11
b) Pr(AUB)=(10+3+12)/33=25/33
c) this is the one I am not 100% sure on. I looked at the set of A and the set of B' seperately then took the intersection, I got Pr(AUB') = (10+3+8)/33 = 21/33

Q2. conditional pr.
Out of a standard deck of 52 cards, 5 cards are chosen at random. What is the probability that the ace of spades is chosen given at least one ace is chosen.

so I'm looking at conditional probability and the formula Pr(A|B) = Pr(A∩B)/Pr(B), where Pr(A) is probability of choosing ace of spades, and Pr(B) is probability of choosing at least one ace.

Pr(B) is simply 1-Pr(no aces).

Is this the right way to go about this question? If so, how do I find pr(no aces) and Pr(A∩B)?

Last edited by a moderator:
t_n_p said:
Q1. Venn diagrams.

[PLAIN]http://img826.imageshack.us/img826/3194/23872542.jpg
Find:
a) Pr(A∩B)
b) Pr(AUB)
c) Pr(AUB')

a) this is easy, Pr(A∩B)=3/33=1/11
b) Pr(AUB)=(10+3+12)/33=25/33
Good!

c) this is the one I am not 100% sure on. I looked at the set of A and the set of B' seperately then took the intersection, I got Pr(AUB') = (10+3+8)/33 = 21/33
No. $$\displaystyle A\cup B'$$ contains 10+ 8= 18 (10= things in A that are not in B, 8= things that are not in A and also not in B). $$\displaystyle Pr(A\cup B')= 18/33= 6/11$$.

Q2. conditional pr.
Out of a standard deck of 52 cards, 5 cards are chosen at random. What is the probability that the ace of spades is chosen given at least one ace is chosen.

so I'm looking at conditional probability and the formula Pr(A|B) = Pr(A∩B)/Pr(B), where Pr(A) is probability of choosing ace of spades, and Pr(B) is probability of choosing at least one ace.

Pr(B) is simply 1-Pr(no aces).

Is this the right way to go about this question? If so, how do I find pr(no aces) and Pr(A∩B)?

Last edited by a moderator:
Ah, so the 3 does not fall under A, but rather in its own group A∩B?
By the same token, then Pr (A'UB) = (12+8)/33?

What about the card problem, any ideas?
Thanks!

## 1. What is a Venn diagram and how is it used in probability?

A Venn diagram is a visual representation of sets and their relationships. In probability, it is used to show the overlap or intersection between different events or outcomes. The areas within the circles represent the elements that are common to both sets, while the areas outside the circles represent elements that are exclusive to each set. This helps to illustrate the probabilities of different outcomes and how they are related.

## 2. How do you calculate conditional probability?

Conditional probability is the probability of an event occurring given that another event has already occurred. It is calculated by dividing the probability of the intersection of the two events by the probability of the first event. For example, if event A has a probability of 0.6 and event B has a probability of 0.4, and the probability of both events occurring together is 0.2, then the conditional probability of event B given event A is P(B|A) = 0.2/0.6 = 0.33.

## 3. What is the difference between mutually exclusive and independent events?

Mutually exclusive events are events that cannot occur at the same time. For example, rolling a 3 and a 5 on a die are mutually exclusive events. Independent events, on the other hand, are events where the outcome of one event does not affect the outcome of the other event. For example, flipping a coin and rolling a die are independent events.

## 4. How can you use Venn diagrams to solve probability problems?

Venn diagrams can be used to visually represent the different events, outcomes, and their relationships in a probability problem. By filling in the overlapping areas with the corresponding probabilities, you can calculate the probabilities of different outcomes and determine the likelihood of certain events occurring. Venn diagrams can also be used to illustrate conditional probability by adjusting the size of the circles and their overlap.

## 5. Can Venn diagrams be used for more complex probability problems?

Yes, Venn diagrams can be used for more complex probability problems by adding more circles and adjusting the size of the overlaps. They can also be used in conjunction with other probability tools, such as tree diagrams, to solve more complicated problems. However, for very complex problems, other approaches such as algebraic methods may be more efficient.

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