Basic Probability - Dealing With Events With Replacement & Independent Events
Hello. This math post for young students is on basic probability with replacement and independent events.
Students should be familiar with multiplying fractions. Knowing exponents is a plus as there is repeated multiplication.
Screenshot images with math text are from my RMarkdown output.
Topics
- Probability With Replacement
- Independent Events
- Practice Problems
- Answers To Practice Problems
Probability With Replacement
One of my previous math posts dealt with probability without replacement when dealing with randomly picking marbles in a bag. When dealing with selecting multiple marbles from the bag if the marbles are not put back in the bag we have sampling without replacement. This changes the number of marbles in the bag and the amount of marbles in the bag for each colour/type.
If all the marbles drawn from the bag are put back, we are dealing with sampling with replacement. The number of marbles in the bag is not changed and so are the number of marbles for each marble colour/type.
Example - Picking Two Marbles From A Bag
In a bag of marbles there are 2 green marbles, 5 red marbles, and 3 black marbles. What is the probability of getting two red marbles if the first marble is not put back in the bag? What is the probability of getting two red marbles if the first marble is put back in the bag?
The probability of selecting a red marble in the first pick is 5 out of 10. If the first red marble is not put back in the bag we have sampling without replacement. The bag has 9 marbles in total in which four of them are red. Obtaining a second red marble has a probability of 4 out of 9. Multiplying the probabilities gives:
Independent Events
In the probability framework, independent events are events of change where one event does not affect the outcome of another event. One example of independent events is the example above where when you put the marble back in the bag, the number of red marbles does not change and the number of marbles in the bag does not change. In the next try the probability of selecting a red marble would stay the same at 5 out of 10 under sampling with replacement.
Another example of independent events is flipping a fair two-sided coin and rolling a die (dice is plural). Flipping a coin does not change the number of sides on a die and rolling a die does not change the number of sides on a coin.
Practice Problems
1) In a two-sided fair coin, what is the probability of flipping a heads three times in all three flips?
2) In a six-sided fair die, what is the probability of rolling a 3 three times in three tries?
3) In a six-sided fair die, what is the probability of rolling a a number 3 or 5 two times in two rolls?
4) What is the probability of flipping a heads of a two-sided fair coin once and rolling a number less than 5 on a six sided die once?
Answers To Practice Problems
Thank you for reading.