In throwing of two dice, the number of exhaustive events that ‘5’ will never appear on any one of the dice is
25
Understanding the Two Dice Throw Problem
The question asks about the number of possible outcomes when two standard six-sided dice are thrown, with the specific condition that the number '5' must not appear on either die. These possible outcomes are often referred to as exhaustive events in probability when considering a specific condition within the total sample space.
Total Outcomes When Throwing Two Dice
When a single standard die is thrown, there are 6 possible outcomes: 1, 2, 3, 4, 5, or 6. When two dice are thrown, the outcome of each die is independent of the other. The total number of possible outcomes in the sample space is the product of the number of outcomes for each die.
Total possible outcomes = Outcomes on Die 1 × Outcomes on Die 2
Total possible outcomes = \(6 \times 6 = 36\)
These 36 outcomes can be represented as ordered pairs (Die 1 result, Die 2 result), for example, (1,1), (1,2), ..., (6,6).
Calculating Outcomes Where '5' Never Appears
The condition is that '5' will never appear on any one of the dice. This means the outcome on the first die cannot be 5, and the outcome on the second die cannot be 5.
Possible Outcomes for a Single Die Without '5'
If the number '5' is excluded, the possible outcomes for a single die are:
- 1
- 2
- 3
- 4
- 6
There are 5 possible outcomes for a single die if '5' is not allowed.
Combining Outcomes for Two Dice Without '5'
Since the outcome on each die must not be '5', we consider the allowed outcomes for each die independently.
- Number of allowed outcomes for the first die (not 5) = 5
- Number of allowed outcomes for the second die (not 5) = 5
To find the total number of outcomes where '5' does not appear on either die, we multiply the number of allowed outcomes for each die.
Number of exhaustive events where '5' never appears = (Allowed outcomes on Die 1) × (Allowed outcomes on Die 2)
Number of exhaustive events = \(5 \times 5\)
Number of exhaustive events = \(25\)
These 25 outcomes include pairs like (1,1), (1,2), ..., (1,6) excluding (1,5); (2,1), ..., (2,6) excluding (2,5), and so on, up to (6,1), ..., (6,6) excluding (6,5), and also excluding the entire row and column corresponding to 5 on either die.
Verification of Outcomes
Let's consider the outcomes where '5' *does* appear on at least one die. These are:
- '5' on the first die: (5,1), (5,2), (5,3), (5,4), (5,5), (5,6) - 6 outcomes.
- '5' on the second die: (1,5), (2,5), (3,5), (4,5), (5,5), (6,5) - 6 outcomes.
The outcome (5,5) is counted in both lists. So, the total number of unique outcomes where '5' appears on at least one die is \(6 + 6 - 1 = 11\).
The number of outcomes where '5' never appears is the total number of outcomes minus the number of outcomes where '5' appears on at least one die.
Total outcomes = 36
Outcomes where '5' appears on at least one die = 11
Outcomes where '5' never appears = \(36 - 11 = 25\).
Both methods yield the same result, confirming that there are 25 exhaustive events where '5' will never appear on any one of the dice when throwing two dice.
Revision Table: Two Dice Probability Concepts
| Concept | Explanation |
|---|---|
| Sample Space | The set of all possible outcomes of an experiment. For two dice, there are 36 outcomes. |
| Exhaustive Events | All possible outcomes of an experiment or all outcomes that satisfy a specific condition within the sample space. |
| Independent Events | Events where the outcome of one does not affect the outcome of the other. Throwing two dice are independent events. |
| Complementary Events | Two events that are mutually exclusive and together cover all possible outcomes. The event "5 appears on at least one die" is complementary to the event "5 never appears on any die". |
Additional Information: Calculating Specific Outcomes
When dealing with probability problems involving multiple independent events like throwing dice, it's often useful to think about the possibilities for each event separately and then combine them. If an event must satisfy condition A AND condition B, and A and B are independent, the total number of outcomes is the number of outcomes satisfying A multiplied by the number of outcomes satisfying B.
In this problem:
- Condition A: The first die does not show a '5'.
- Condition B: The second die does not show a '5'.
Number of outcomes for Condition A = 5 (any number except 5).
Number of outcomes for Condition B = 5 (any number except 5).
Total outcomes satisfying both A and B = \(5 \times 5 = 25\).
This method is generally simpler when the condition is about excluding a specific outcome from each independent trial.
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