Two dice are thrown simultaneously and the sum of the numbers appearing on them is noted. What is the probability that the sum is 12?
Calculating Probability When Two Dice Are Thrown
This problem asks for the probability of getting a specific sum when two standard six-sided dice are thrown simultaneously. To solve this, we first need to understand the total possible outcomes and the number of outcomes that satisfy the condition (the sum is 12).
Understanding the Sample Space of Throwing Two Dice
When you throw one standard die, there are 6 possible outcomes: 1, 2, 3, 4, 5, or 6. When you throw two dice, the outcome is a pair of numbers, one from each die. The total number of possible outcomes is the product of the outcomes for each die.
Total number of outcomes = (Outcomes for die 1) $\times$ (Outcomes for die 2)
Total number of outcomes = $6 \times 6 = 36$
We can list all these possible outcomes as pairs (result on die 1, result on die 2). This list of all possible outcomes is called the sample space.
| Die 1 \ Die 2 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |
| 2 | (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |
| 3 | (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |
| 4 | (4,1) | (4,2) | (4,3) | (4,4) | (4,5) | (4,6) |
| 5 | (5,1) | (5,2) | (5,3) | (5,4) | (5,5) | (5,6) |
| 6 | (6,1) | (6,2) | (6,3) | (6,4) | (6,5) | (6,6) |
Identifying Favourable Outcomes for Sum 12
The problem asks for the probability that the sum of the numbers appearing on the two dice is 12. We need to look at the sample space and find all the pairs where the two numbers add up to 12.
Let's check the sums for each pair:
- Pairs with sum 2: (1,1)
- Pairs with sum 3: (1,2), (2,1)
- ...
- Pairs with sum 11: (5,6), (6,5)
- Pairs with sum 12: (6,6)
The only outcome where the sum of the numbers is exactly 12 is the pair (6,6).
Number of favourable outcomes (sum is 12) = 1
Calculating the Probability
The probability of an event is calculated using the formula:
Probability (Event) = $\frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$
In this case, the event is "the sum of the numbers is 12".
- Number of favourable outcomes (sum is 12) = 1
- Total number of possible outcomes = 36
So, the probability is:
Probability (Sum is 12) = $\frac{1}{36}$
Therefore, the probability that the sum of the numbers appearing on the two dice is 12 is $\frac{1}{36}$.
Revision Table: Key Probability Concepts
| Concept | Definition | Example (Two Dice) |
|---|---|---|
| Experiment | A process that results in one of several possible outcomes. | Throwing two dice |
| Outcome | A single possible result of an experiment. | Getting (3, 4) when throwing two dice |
| Sample Space (S) | The set of all possible outcomes of an experiment. | The 36 pairs listed above |
| Event (E) | A subset of the sample space; a collection of one or more outcomes. | Getting a sum of 12 |
| Favourable Outcome | An outcome that satisfies the condition of the event. | The outcome (6,6) for the event "sum is 12" |
| Probability (P(E)) | The likelihood of an event occurring. Calculated as $\frac{\text{Number of Favourable Outcomes}}{\text{Total Outcomes}}$. | $P(\text{Sum is 12}) = \frac{1}{36}$ |
Additional Information: Other Dice Probabilities
We calculated the probability of getting a sum of 12 with two dice. Let's briefly look at probabilities for other sums to better understand the distribution:
- Smallest possible sum: 1 + 1 = 2 (Outcome: (1,1)) - Probability = $\frac{1}{36}$
- Largest possible sum: 6 + 6 = 12 (Outcome: (6,6)) - Probability = $\frac{1}{36}$
- Most frequent sum: 7 (Outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)) - Probability = $\frac{6}{36} = \frac{1}{6}$
As you can see, sums in the middle of the range (like 7) are more likely because there are more combinations of numbers that add up to them, compared to the extreme sums (like 2 or 12).
Important Questions from Probability
If a box contains 3 white cushions, 4 red cushions and 5 blue cushions, what is the probability of selecting a white or blue cushion?
A. 2/3
B. 3/4
C. 1/4
D. 1/9Statements followed by some conclusions are given below.
Statements:
1. A bag has 2 white, 3 black, 4 red and 6 green balls.
2. 1 ball selected at random from the bag.
Conclusions:
I. The probability that a black ball is selected is 1/5
II. The probability that a red ball is selected is 6/15
Find which of the conclusions logically follows from the given statement
A. Only conclusion I follows.
B. Only conclusion II follows.
C. Both I and II follow.
D. Neither I nor II follows.
In a shooting test, the probabilities of hitting the target are 1/2 for A, 2/3 for B and 3/4 for C. If they fire at the same target, what is the probability that only one of them hits the target?
A bag contains balls numbered from 1 to 42. One ball is drawn at random from these balls. The probability that its number is a multiple of 7 or 8 is:
A bag contains $3$ red balls, $4$ white balls, and $5$ green balls. Of these, three balls are drawn at random. What is the probability that exactly two of the balls drawn are of the same colour?
