James has 2 quarters, 3 dimes, and 7 nickels in his pocket. He pulls a coin out of his pocket and then pulls out another coin without replacing the first coin.
- Let A be the event : "James gets a dime on the first pull".
- Let B be the event : "James gets a nickel on the second pull".
Are A and B independent?
Two events are independent if the result of the second event is not affected by the result of the first event.
Here, the size of the sample is 12. Since James does not replace the first coin, the size of the sample changes to 11. Therefore, the events are dependent.
No, A and B are dependent.
Emma rolls a fair die twice.
- Let A be the event : "Emma gets an even number on the first roll".
- Let B be the event : "The sum the two numbers that Emma gets is 5".
Are A and B independent?
Two events are independent if the result of the second event is not affected by the result of the first event.
Here the result of event A will affect event B. For example, if she rolls a 6 on the first roll, then the probability of event B would be zero. If she gets 2 on the first roll and 3 on the second roll, than event B will occur (the probability of event B occurring in this case is \dfrac{1}{6} ).
Therefore, the events are dependent.
No, A and B are dependent.
Alice and Sarah are competing in an archery game. According to their previous records, the probability that Alice and Sarah will hit the target is 0.6 and 0.7 respectively. Each player can shoot 5 times.
- Let A be the event : "Alice hits the target 3 times".
- Let B be the event : "Sarah hits the target 4 times".
Are A and B independent?
Two events are independent if the result of the second event is not affected by the result of the first event.
Here, we can see that the number of successful shoots of Alice would not affect the result of Sarah. Therefore, the events are independent.
Yes, A and B are independent.
In a family with two children, consider the following events:
- The event : "The first child is a girl".
- The event : "The second child is a boy".
Are A and B independent?
Two events are independent if the result of the second event is not affected by the result of the first event.
Here, the gender of the first child does not affect the gender of the second child. Therefore, the events are independent.
Yes, A and B are independent.
We flip a coin three times.
- Let A be the event : "We get at least two tails".
- Let B be the event : "We get at least one head".
Are A and B independent?
Two events are independent if the result of the second event is not affected by the result of the first event.
Here, the size of the sample space is 8. The probability that event B occurs is equal to \dfrac{7}{8} (the only unwilling case is getting three tails whose probability is \dfrac{1}{8} ). Now, assuming that event A has happened, the size of the sample space changes to 4:
\left(h,t,t\right), \left(t,h,t\right),\left(t,t,h\right), \left(t,t,t\right)
The probability of event B will change to \dfrac{3}{4}.
Event B would be affected by event A.
No, A and B are dependent.
In a standard 52 card deck, we draw two cards simultaneously.
- Let A be the event : "both cards are hearts".
- Let B be the event : "both cards are queens".
Are A and B independent?
Two events are independent if the result of the second event is not affected by the result of the first event.
Here, if the event A happens, then the probability of event B occurring will change to zero. Therefore, the events are dependent.
No, A and B are dependent.
We roll a die twice.
- Let A be the event : "the first number we get is even".
- Let B be the event : "the second number we get is 6 ".
Are A and B independent?
Two events are independent if the result of the second event is not affected by the result of the first event.
In this question, the probability that we get 6 on the second roll is not affected by the result of the first roll. Therefore, the events are independent.
Yes, A and B are independent.