Use the three formulas above to answer the following questions.
What is the probability of drawing a
What is the probability of rolling an odd number on a standard
If the event
If the event
The probability of drawing a
The probability of rolling an odd number is
Since
Since
Use DeMorgan’s Laws to solve the following questions.
If the probability of the intersection of
If the probability of the intersection of the complement of
We can rewrite this question as: if
We can rewrite this question as: if
Determine whether the following events are disjoint or not. If the events are disjoint, give the probability of
A coin is flipped with the events:
A coin is flipped twice with the events:
A card is drawn from a standard deck of
It is impossible for a coin flip to be both Heads and Tails at the same time, so the events
which makes sense because
These events are not disjoint because it is possible to get Heads on the first coin flip and then get Tails on the second coin flip.
It is impossible for a card to be both a 10 and a face-card, so the events
Determine if the following events are independent or dependent. If they are independent then solve for the probability of the intersection,
When flipping a coin,
When flipping a coin,
When rolling a 6-sided die,
In a student council election, there are
The outcomes of different coin flips do not affect one another, so the events
The outcomes of different coin flips do not affect one another, so the events
Since the events are for the same dice-roll, instead of different dice-rolls, the event
The outcome of either position will result in
Suppose we flip a coin three times.
What is the probability of getting Heads on the first coin flip?
What is the probability of getting Tails on the third coin flip?
What is the probability of not getting Tails on the second coin flip?
What is the probability of getting Heads on all three coin flips?
Solution:
The probability of getting Heads for any coin flip is
The probability of getting Tails for any coin flip is
The probability of not getting Tails for any coin flip is the same as getting Heads for any coin flip, which is
We can define the events
A sack contains marbles of different colours. There are
How many red marbles are in the sack?
How many marbles in the sack are not red?
What is the probability of drawing a marble that is not red?
Solution:
We define the event
Thus, there are
We have that
Thus, there are 14 marbles in the sack that are not red.
There are two ways we can solve this problem.
(1) From part (b), we have that
Thus, the probability of drawing a marble that is not red is
(2) Another way we could have solved this is using one of the formulas for the probability of the complement of an event. Since
Thus, the probability of drawing a marble that is not red is
An aquarium contains two kinds of fish: clownfish and pufferfish. Let the event
How many fish are there in total?
What is
Use any method to determine
Solution:
We wish to find
Thus, there are 150 fish in total in the aquarium.
Since there are only two kinds of fish in the aquarium, instead of just saying
There are two ways we can solve this problem.
(1) From parts (a) and (b), we have that
(2) From the question and part (a), we have
Use the formulas for the relationship between the probabilities of the union and intersection of events to answer the following.
Determine
Determine
Determine
Solution:
TRUE or FALSE: Is it possible to have the following probabilities? Justify your answer.
Solution: No, it is not possible to have these probabilities. We must have that
but this is not true here since
At an exclusive social event, each guest is given a wristband that is either blue, green, orange or yellow. Of the
Determine the probability of each of the events.
Are the events is disjoint or not? Justify your answer.
What is the probability that a randomly selected person has either a blue wristband or a yellow wristband?
What is the probability that a randomly selected person doesn’t have a blue wristband and doesn’t have a yellow wristband?
Solution:
From the question, we have
The events
We wish to find
Thus, there is a 0.518 probability that a randomly selected person has either a blue wristband or a yellow wristband.
We wish to find
(1) The first way to solve this is by using the result from part (c) and DeMorgan’s Laws. From part (c), we have that
We then use one of DeMorgan’s Laws to get:
Thus, there is a
(2) Another way to solve this is by realizing that a person not having a blue wristband and not having yellow wristband just means that they have either a green wristband or orange wristband. So,
Then, since the events are all disjoint, we have that:
Thus, there is a
In a candy jar, the candy is categorized by the following attributes that have no influence on one another: the candy is either sweet or sour; and the candy is either hard or soft. The events are
Determine
Are the events
Determine
What is the sum of the probabilities from part (c)? Why is this the case?
Solution:
It is explicitly stated in the question that the attributes have no influence on one another. That is, whether a candy is sweet or sour doesn’t affect if the candy is hard or soft. So, the events
Since the events are all independent, we have:
The sum of these probabilities is 1 because these are all the possible combinations we can have for the candy; soft and sweet, soft and sour, hard and sweet, or hard and sour.
Is it possible for two events
Solution: Let us assume that two events
Note: that if either
Suppose you are on a game show and are presented with
Solution: Initially, there are
Let us label the doors as
Now, even more doors with nothing behind them are removed until only