Paul throws an unbiased die.
Let A be the event : "the result is even".
Let B be the event : "the result of the die is greater than or equal to 3".
Find P\left(A\cap B\right).
As all outcomes are equally likely, the probability of an event E is defined as the number of outcomes favorable to the event E divided by the total number of possible outcomes.
Number of possible outcomes
Paul is throwing a unbiased die. The possible outcomes are:
\left\{ 1, 2, 3, 4, 5, 6 \right\}
Therefore, the total number of possible outcomes is 6.
Number of favorable outcomes
Since we want both events A and B to occur, the die must give an even number and greater or equal to 3. The favorable outcomes are:
\left\{ 4, 6 \right\}
Therefore, the number of favorable outcomes is 2.
Conclusion
The probability is:
P\left(A\cap B\right)=\dfrac{2}{6}
P\left(A\cap B\right)=\dfrac13
In a bag there are 5 red marbles, 10 blue marbles, 15 green marbles, and 20 yellow marbles. Joey picks a marble.
What is the probability of Joey picking a blue marble?
Let E be the event "Joey pick a blue marble".
As all outcomes are equally likely, the probability of an event E is defined as the number of outcomes favorable to the event E divided by the total number of possible outcomes.
Number of possible outcomes
The total number of possible outcomes is the total number of marbles in the bag:
5+10+15+20=50
Number of favorable outcomes
Since Joey wants to pick a blue marble, the number of favorable outcomes is the number of blue marbles: 10.
Conclusion
The probability is:
P\left(E\right)=\dfrac{10}{50}=\dfrac{1}{5}.
P\left(E\right)=\dfrac{1}{5}
In a bag there are 10 red cards numbered 1 through 10 and 7 blue cards numbered 1 through 7.
Bob is picking a card.
Let A be the event: "the color of the card is red."
Let B be the event: "the number on the card is greater than 6."
Find P\left(A\cap B\right).
As all outcomes are equally likely, the probability of an event E is defined as the number of outcomes favorable to the event E divided by the total number of possible outcomes.
Number of possible outcomes
The total number of possible outcomes is the total number of cards in the bag:
7+10=17
Number of favorable outcomes
Since we want both events A and B to occur, the color of the card must be red and the number on the card must be greater than 6. The favorable outcomes are:
\left\{ Red7, Red8, Red9, Red10\right\}
Therefore, the number of favorable outcomes is 4.
Conclusion
The probability is:
P\left(A\cap B\right)=\dfrac{4}{17}
P\left(A\cap B\right)=\dfrac{4}{17}
At a local car dealership there are 20 new trucks and 19 new cars. Also, at the same dealership there are 5 second-hand trucks and 16 second-hand cars.
Determine the probability that a randomly selected vehicle is new.
Let E be the event "The selected vehicle is new".
As all outcomes are equally likely, the probability of an event E is defined as the number of outcomes favorable to the event E divided by the total number of possible outcomes.
Number of possible outcomes
The total number of possible outcomes is the total number of vehicles at the local dealership:
20+19+5+16=60
Number of favorable outcomes
The number of favorable outcomes is the number of new vehicles:
20+19=39
Conclusion
The probability is:
P\left(E\right)=\dfrac{39}{60}=\dfrac{13}{20}.
P\left(E\right)=\dfrac{13}{20}
Mark throws an unbiased die.
Let A be the event : "the result is odd".
Let B be the event : "the result of the die is greater than or equal to 4".
Find P\left(A\cup B\right).
As all outcomes are equally likely, the probability of an event E is defined as the number of outcomes favorable to the event E divided by the total number of possible outcomes.
Number of possible outcomes
Mark is throwing a unbiased die. The possible outcomes are:
\left\{ 1, 2, 3, 4, 5, 6 \right\}
Therefore, the total number of possible outcomes is 6.
Number of favorable outcomes
For the event A, the favorable outcomes are:
\left\{ 1, 3, 5 \right\}
In conclusion, the probability of event A is:
P\left(A\right)=\dfrac{3}{6}=\dfrac{1}{2}
For the event B, the favorable outcomes are:
\left\{ 4, 5, 6 \right\}
In conclusion, the probability of event B is:
P\left(B\right)=\dfrac{3}{6}=\dfrac{1}{2}
For the A\cap B, we want both events A and B to occur, so the dice must give an odd number greater than or equal to 4. The only favorable outcome is:
\left\{ 5 \right\}
Therefore the number of favorable outcomes is 1.
In conclusion, the probability of event A\cap B is:
P\left(A\cap B\right)=\dfrac{1}{6}
Conclusion
The formula to calculate the probability of a union is:
P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)
The final probability is:
P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)=\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{6}=\dfrac{5}{6}.
P\left(A\cup B\right)=\dfrac{5}{6}
In a bag there are 12 green cards numbered 1 through 12 and 10 yellow cards numbered 1 through 10.
Luke is picking a card.
Let A be the event: "the color of the card is yellow."
Let B be the event: "the number on the card is greater than 8."
Find P\left(A\cap B\right).
Number of possible outcomes
As all outcomes are equally likely, the probability of an event E is defined as the number of outcomes favorable to the event E divided by the total number of possible outcomes.
The total number of possible outcomes is the total number of cards in the bag:
12+10=22
Number of favorable outcomes
Since we want both events A and B to occur, the color of the card must be yellow and the number on the card must be greater than 8. The favorable outcomes are:
\left\{ Yellow9, Yellow10 \right\}
Therefore, the number of favorable outcomes is 2.
Conclusion
The probability is:
P\left(A\cap B\right)=\dfrac{2}{22}=\dfrac{1}{11}.
P\left(A\cap B\right)=\dfrac{1}{11}