Problem 11 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

The odds against throwing 7 with 2 dice in a throw are: (a) \(5: 1\) (b) \(1: 5\) (c) \(1: 4\) (d) \(3: 1\)

Step-by-Step Solution

Answer

The odds against throwing 7 with 2 dice in a throw are \(5:1\).

Step 1: Calculate Total Possible Outcomes

When two dice are thrown, each die has 6 faces. Therefore, the total number of possible outcomes is given by multiplying the number of outcomes for each die: \[ \text{Total Outcomes} = 6 \times 6 = 36 \]

Step 2: Determine Favorable Outcomes for a Sum of 7

We need to find the combinations on two dice that sum to 7. These combinations are: - (1,6) - (2,5) - (3,4) - (4,3) - (5,2) - (6,1) This gives us a total of 6 favorable outcomes.

Step 3: Calculate Unfavorable Outcomes

Since we have 36 possible outcomes in total and 6 favorable outcomes for getting a sum of 7, the number of unfavorable outcomes is:\[ \text{Unfavorable Outcomes} = 36 - 6 = 30 \]

Step 4: Find the Odds Against Throwing 7

The odds against an event occurring are given by the ratio of unfavorable outcomes to favorable outcomes. Thus, the odds against throwing a 7 are:\[ \text{Odds Against} = \frac{\text{Unfavorable Outcomes}}{\text{Favorable Outcomes}} = \frac{30}{6} = 5:1 \]

Key Concepts

Dice Combinations

When rolling two dice, each die has six faces, making it possible to create a variety of combinations. Each die has faces numbered 1 through 6. When rolled together, the dice can produce different sums depending on the numbers that appear face up.
Understanding dice combinations involves calculating all possible outcomes. For two dice, since each has 6 faces, you end up with a total of 36 unique combinations. This is because each face on the first die can pair with each face on the second die, calculated as:

• Total Possible Outcomes = 6 faces × 6 faces = 36 combinations

These combinations include pairs like (1,1), (1,2), up to (6,6). When seeking specific outcomes, such as a sum of 7, only particular combinations work. The desirable pairs are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), totaling 6 combinations.
This concept is fundamental in probability as it sets the base for determining the odds of rolling any specific sum.

Odds and Events

Odds offer a unique way to discuss probability, different from probability itself. The odds against a particular event detail how likely it is not to happen, compared to it happening. Consider rolling a 7 with two dice, which has been calculated, gives us understanding of favorable and unfavorable outcomes.
By counting, we found 6 ways to roll a sum of 7. Conversely, the unfavorable outcomes, which are the non-7 sums, consist of the remaining combinations. From the total 36 possibilities, subtracting 6 favorable outcomes leaves us with 30 unfavorable outcomes.
Expressing odds involves forming a ratio of unfavorable events to favorable events:

• Odds Against = Unfavorable Outcomes : Favorable Outcomes = 30 : 6
• Which simplifies to 5 : 1

This ratio tells us that for every 6 times we roll a dice pair, we can "expect" not to get a 7 around 5 of those times, with a better understanding of what makes outcomes more or less likely.

Permutations and Combinations

In probability, permutations and combinations explain the ways to arrange or select items. Both concepts are crucial in understanding outcomes, especially in games involving dice.
Permutations focus on sequences. When order matters, permutations help in counting distinct arrangements. However, in dice rolls like summing to 7, order doesn't affect probabilities (a result like (3,4) is equal to (4,3)). Combinations become valuable when order doesn’t matter. With dice, we're interested in the count of certain sums occurring, rather than "different" orders for each sum:

• Example: The viable combinations for a sum of 7 involve unordered pairs like (1,6) and (6,1) being equal.

Understanding how to count events correctly with combinations helps in deducing the feasible outcomes or favorable events among the total possible outcomes. This understanding aids in calculating probabilities accurately, pivotal in dice-based games and more complex statistical analyses.