A Fair Coin And A Fair Dye Are Thrown Together Once Using A Table Of Possible Outcomes Find The Probability Of Obtaining A Prime Number And The Two , No Tail And No Multiple Of 3
A Fair Coin and a Fair Die: Exploring Probabilities
In probability theory, we often use simple experiments to understand the concept of chance and uncertainty. One such experiment involves throwing a fair coin and a fair die together. In this article, we will explore the possible outcomes of this experiment and calculate the probability of obtaining a prime number and the two, no tail, and no multiple of 3.
Let's consider the possible outcomes of throwing a fair coin and a fair die together. A fair coin has two possible outcomes: Head (H) or Tail (T). A fair die has six possible outcomes: 1, 2, 3, 4, 5, or 6. When we throw the coin and the die together, we have a total of 12 possible outcomes:
| Coin | Die | Outcome |
|---|---|---|
| H | 1 | H1 |
| H | 2 | H2 |
| H | 3 | H3 |
| H | 4 | H4 |
| H | 5 | H5 |
| H | 6 | H6 |
| T | 1 | T1 |
| T | 2 | T2 |
| T | 3 | T3 |
| T | 4 | T4 |
| T | 5 | T5 |
| T | 6 | T6 |
A prime number is a positive integer that is divisible only by itself and 1. In the context of this experiment, we are interested in the outcomes that result in a prime number. The prime numbers between 1 and 6 are 2, 3, and 5.
| Coin | Die | Outcome | Prime |
|---|---|---|---|
| H | 2 | H2 | Yes |
| H | 3 | H3 | Yes |
| H | 5 | H5 | Yes |
| T | 2 | T2 | Yes |
| T | 3 | T3 | Yes |
| T | 5 | T5 | Yes |
There are 6 outcomes that result in a prime number.
We are also interested in the outcomes that do not result in a Tail. The outcomes that do not result in a Tail are:
| Coin | Die | Outcome |
|---|---|---|
| H | 1 | H1 |
| H | 2 | H2 |
| H | 3 | H3 |
| H | 4 | H4 |
| H | 5 | H5 |
| H | 6 | H6 |
There are 6 outcomes that do not result in a Tail.
We are also interested in the outcomes that do not result in a multiple of 3. The outcomes that do not result in a multiple of 3 are:
| Coin | Die | Outcome |
|---|---|---|
| H | 1 | H1 |
| H | 2 | H2 |
| H | 4 | H4 |
| H | 5 | H5 |
| T | 1 | T1 |
| T | 2 | T2 |
| T | 4 | T4 |
| T | 5 | T5 |
There are 8 outcomes that do not result in a multiple of 3.
Now that we have identified the outcomes of interest, we can calculate the probabilities.
Probability of Obtaining a Prime Number
The probability of obtaining a prime number is the number of outcomes that result in a prime number divided by the total number of outcomes.
Probability of obtaining a prime number = 6/12 = 1/2
Probability of No Tail
The probability of no Tail is the number of outcomes that do not result in a Tail divided by the total number of outcomes.
Probability of no Tail = 6/12 = 1/2
Probability of No Multiple of 3
The probability of no multiple of 3 is the number of outcomes that do not result in a multiple of 3 divided by the total number of outcomes.
Probability of no multiple of 3 = 8/12 = 2/3
Probability of Obtaining a Prime Number and No Tail
The probability of obtaining a prime number and no Tail is the number of outcomes that result in a prime number and no Tail divided by the total number of outcomes.
Probability of obtaining a prime number and no Tail = 3/12 = 1/4
Probability of Obtaining a Prime Number and No Multiple of 3
The probability of obtaining a prime number and no multiple of 3 is the number of outcomes that result in a prime number and no multiple of 3 divided by the total number of outcomes.
Probability of obtaining a prime number and no multiple of 3 = 3/12 = 1/4
In this article, we explored the possible outcomes of throwing a fair coin and a fair die together. We calculated the probabilities of obtaining a prime number, no Tail, and no multiple of 3. We also calculated the probabilities of obtaining a prime number and no Tail, and a prime number and no multiple of 3. The results show that the probability of obtaining a prime number is 1/2, the probability of no Tail is 1/2, and the probability of no multiple of 3 is 2/3. The probability of obtaining a prime number and no Tail is 1/4, and the probability of obtaining a prime number and no multiple of 3 is 1/4.
- [1] Probability Theory, by E.T. Jaynes
- [2] Statistics, by W.H. Kruskal and J.M. Tanur
- [3] Probability and Statistics, by J.L. Devore
- Prime number: A positive integer that is divisible only by itself and 1.
- Fair coin: A coin that has two possible outcomes: Head (H) or Tail (T).
- Fair die: A die that has six possible outcomes: 1, 2, 3, 4, 5, or 6.
- Probability: A measure of the likelihood of an event occurring.
A Fair Coin and a Fair Die: Q&A
In our previous article, we explored the possible outcomes of throwing a fair coin and a fair die together. We calculated the probabilities of obtaining a prime number, no Tail, and no multiple of 3. In this article, we will answer some frequently asked questions related to this experiment.
Q: What is the probability of obtaining a prime number and no Tail?
A: The probability of obtaining a prime number and no Tail is 1/4. This is because there are 3 outcomes that result in a prime number and no Tail (H2, H3, and H5), and a total of 12 possible outcomes.
Q: What is the probability of obtaining a prime number and no multiple of 3?
A: The probability of obtaining a prime number and no multiple of 3 is 1/4. This is because there are 3 outcomes that result in a prime number and no multiple of 3 (H2, H5, and T2), and a total of 12 possible outcomes.
Q: What is the probability of obtaining a prime number, no Tail, and no multiple of 3?
A: The probability of obtaining a prime number, no Tail, and no multiple of 3 is 1/12. This is because there are 3 outcomes that result in a prime number, no Tail, and no multiple of 3 (H2, H5, and T2), and a total of 12 possible outcomes.
Q: What is the probability of obtaining a multiple of 3 and a Tail?
A: The probability of obtaining a multiple of 3 and a Tail is 2/12 = 1/6. This is because there are 2 outcomes that result in a multiple of 3 and a Tail (T3 and T6), and a total of 12 possible outcomes.
Q: What is the probability of obtaining a multiple of 3 and no Tail?
A: The probability of obtaining a multiple of 3 and no Tail is 4/12 = 1/3. This is because there are 4 outcomes that result in a multiple of 3 and no Tail (H3, H6, T3, and T6), and a total of 12 possible outcomes.
Q: What is the probability of obtaining a multiple of 3 and a prime number?
A: The probability of obtaining a multiple of 3 and a prime number is 0. This is because there are no outcomes that result in a multiple of 3 and a prime number.
Q: What is the probability of obtaining a multiple of 3, no Tail, and no multiple of 3?
A: The probability of obtaining a multiple of 3, no Tail, and no multiple of 3 is 0. This is because there are no outcomes that result in a multiple of 3, no Tail, and no multiple of 3.
In this article, we answered some frequently asked questions related to the experiment of throwing a fair coin and a fair die together. We calculated the probabilities of obtaining a prime number, no Tail, and no multiple of 3, as well as the probabilities of obtaining a multiple of 3 and a Tail, a multiple of 3 and no Tail, and a multiple of 3 and a prime number.
- [1] Probability Theory, by E.T. Jaynes
- [2] Statistics, by W.H. Kruskal and J.M. Tanur
- [3] Probability and Statistics, by J.L. Devore
- Prime number: A positive integer that is divisible only by itself and 1.
- Fair coin: A coin that has two possible outcomes: Head (H) or Tail (T).
- Fair die: A die that has six possible outcomes: 1, 2, 3, 4, 5, or 6.
- Probability: A measure of the likelihood of an event occurring.