Help finding probability involving combinations?
What is the probability of winning in this game ( Show calculation and all works)
- The board is a number ranging from 1 - 12
- Player place a coin in the desire number 1,2,3,4 etc.
- Roll 2 dice
- To win get the sum of the two number, or the individual number from each die.
For example if the player placed a coin in the number 5.
Then rolled the two dice, the two number that appeared are 5 and 3.
To win player need to placed the coin in 3 or 5 or 8 (sum of 5 and 3)
In this case the player won.
Again what is the probability of winning this game, please show solution. Permutation or Combination?
To find the probability of winning in this game, we need to calculate the number of favorable outcomes (winning outcomes) divided by the total number of possible outcomes.
1. Total number of possible outcomes:
Since there are two dice, each with 6 faces, the total number of outcomes when rolling the two dice is 6 * 6 = 36.
2. Number of favorable outcomes:
To calculate the number of favorable outcomes, we need to consider each scenario separately.
Scenario 1: Winning by getting the sum of two numbers:
There are multiple combinations that can result in the same sum. For each possible sum from 2 to 12, we need to find the combinations of two numbers on the dice that yield that sum.
For example, to get a sum of 7, the possible combinations are (1, 6), (6, 1), (2, 5), (5, 2), (3, 4), and (4, 3). Thus, there are 6 possible outcomes leading to a sum of 7.
Calculating this for all possible sums, we find the following outcomes:
Sum 2: 1 outcome
Sum 3: 2 outcomes
Sum 4: 3 outcomes
Sum 5: 4 outcomes
Sum 6: 5 outcomes
Sum 7: 6 outcomes
Sum 8: 5 outcomes
Sum 9: 4 outcomes
Sum 10: 3 outcomes
Sum 11: 2 outcomes
Sum 12: 1 outcome
Adding up all the outcomes, we get:
1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36
Scenario 2: Winning by getting one of the individual numbers from each die:
For each number on the first die, there are 6 options for the number on the second die. So there are 6 outcomes for each number on the first die, giving us a total of 6 * 6 = 36 outcomes.
3. Number of favorable outcomes:
Adding up the outcomes from both scenarios, we get a total of 36 + 36 = 72 favorable outcomes.
4. Probability of winning:
The probability of winning is the number of favorable outcomes divided by the total number of possible outcomes.
So, the probability of winning in this game is 72/36 = 2/1 or 2 (since there is a 1 in 2 chance of winning).
Therefore, the probability of winning this game is 2/1 or 2 (50%).
To find the probability of winning in this game, we can use the concept of combinations.
Let's break down the problem step by step:
1. The board has numbers ranging from 1 to 12. So, there are 12 possible numbers where the player can place the coin.
2. When rolling two dice, there are a total of 6 possible outcomes for each die (numbers 1 to 6).
3. To win, the player needs to either match the sum of the two numbers rolled or match one of the individual numbers.
To calculate the probability, we need to find the number of favorable outcomes (ways to win) and the total possible outcomes.
Number of favorable outcomes:
- To win by matching the sum of the two numbers, the player needs to place the coin on that sum. There are 11 possible sums from 2 to 12.
- To win by matching one of the individual numbers, the player needs to place the coin on the number itself. Since there are 12 numbers on the board, there are 12 favorable outcomes in this case.
Total possible outcomes:
- Rolling two dice generates all possible combinations of numbers on the dice. This can be found using the concept of permutations. Since each die has 6 possible outcomes, the total number of outcomes is 6 × 6 = 36.
Now we can calculate the probability of winning:
Total probability = Number of favorable outcomes / Total possible outcomes
Total probability = (11 favorable outcomes + 12 favorable outcomes) / 36 total outcomes
Total probability = 23 / 36
So, the probability of winning this game is 23/36, or approximately 0.639.