An octahedral die has eight faces numbered 1 to 8, each of which comes up with equal likelihood (you can see a picture of one in Diagram 3 of the Test 1 Supplement).
Two such dice are rolled, and the product of the numbers is noted. In lowest terms, what are the odds against the product being greater than or equal to 40? Justify your answer.
We can use complementary probability. If we consider a single die, a throw of 5 or higher is required for the product of the two dice to be at least 20, and a throw of 8 is required for the product to exceed 40. Thus, to make the numbers slightly easier, we consider the probability that the product of the two numbers is less than 40. We tabulate the conditions for product less than 40 with different styles denoting simultaneous conditions when necessary:
\[\]
\[
\begin{array}{c||c|c|c|c}
\text{First Die} & \multicolumn{4}{c}{\text{Second Die}} \\
\hline
1,2 & \text{All} & - & - & - \\
\hline
3 & 1\text{-}7,\textbf{\textit{8}} & 1\text{-}3,\textbf{\textit{4}} & 1\text{-}2,\textit{\text{5}} & - \\
\hline
4 & 1\text{-}6,\textbf{\textit{7}} & 1\text{-}4,\textbf{\textit{5}} & 1\text{-}2,\textit{\text{6}} & - \\
\hline
5 & 1\text{-}5,\textbf{\textit{6}} & 1\text{-}3,\textbf{\textit{7}} & \text{All on }\textit{8} & - \\
\hline
6,\textbf{\textit{8}} & \text{All} & \text{All} & \text{All} & - \\
\end{array}
\]
\[\]
There are $\text{7, 4, 5, 6, 5, 7}$ multiplicities for $\text{First Die throws 1-6}$, so the probability is $\frac{7+4+5+6+5+7}{48} = \frac{34}{48}$ meaning an odds of $\boxed{\frac{7}{17}}$.
To find the odds against the product being greater than or equal to 40 when two octahedral dice are rolled, we need to determine the favorable outcomes (where the product is less than 40) and the total possible outcomes.
Step 1: Determine the favorable outcomes
To find the favorable outcomes, we can calculate all the possible combinations of two dice rolls where the product is less than 40.
The possible combinations whose product is less than 40 are:
1 * 1 = 1
1 * 2 = 2
1 * 3 = 3
1 * 4 = 4
1 * 5 = 5
1 * 6 = 6
2 * 1 = 2
2 * 2 = 4
2 * 3 = 6
2 * 4 = 8
2 * 5 = 10
2 * 6 = 12
3 * 1 = 3
3 * 2 = 6
3 * 3 = 9
3 * 4 = 12
4 * 1 = 4
4 * 2 = 8
4 * 3 = 12
5 * 1 = 5
5 * 2 = 10
6 * 1 = 6
There are 22 favorable outcomes where the product is less than 40.
Step 2: Determine the total possible outcomes
To determine the total possible outcomes, we need to calculate all the possible combinations of two dice rolls.
For each dice roll, there are 8 possible outcomes (numbers 1 to 8). Since we are rolling two dice, the total number of outcomes is 8 * 8 = 64.
Step 3: Calculate the odds against the product being greater than or equal to 40
The odds against the product being greater than or equal to 40 can be calculated by dividing the number of unfavorable outcomes (total possible outcomes - favorable outcomes) by the total number of outcomes.
Unfavorable outcomes = Total possible outcomes - Favorable outcomes
Unfavorable outcomes = 64 - 22 = 42
Therefore, the odds against the product being greater than or equal to 40 are 42:22 or simplified as 21:11.
Justification:
The odds can be justified by considering the concept of probability. Probability is calculated by dividing the number of favorable outcomes by the total number of outcomes.
Here, we can see that out of the 64 possible outcomes, 22 outcomes have a product less than 40. Thus, the probability of the product being less than 40 is 22/64.
The odds against an event can be calculated by subtracting the probability of the event occurring from 1 and expressing it as a ratio of unfavorable to favorable outcomes. In this case, the odds against the product being greater than or equal to 40 can be calculated as (1 - 22/64) = 42/64, which simplifies to 21:11.
To find the odds against the product of two dice being greater than or equal to 40, we need to first determine the total number of possible outcomes and then calculate the number of favorable outcomes.
Step 1: Determine the total number of possible outcomes
Each die has 8 possible outcomes (numbers 1 to 8), and since two dice are being rolled, the total number of outcomes is 8 * 8 = 64.
Step 2: Calculate the number of favorable outcomes
We want to find the product of two dice rolls that is greater than or equal to 40. We need to find all the combinations that satisfy this condition.
Let's list all the possible products and determine if they meet the condition:
1 * 1 = 1 (less than 40)
1 * 2 = 2 (less than 40)
1 * 3 = 3 (less than 40)
1 * 4 = 4 (less than 40)
1 * 5 = 5 (less than 40)
1 * 6 = 6 (less than 40)
1 * 7 = 7 (less than 40)
1 * 8 = 8 (less than 40)
2 * 1 = 2 (less than 40)
2 * 2 = 4 (less than 40)
2 * 3 = 6 (less than 40)
2 * 4 = 8 (less than 40)
2 * 5 = 10 (less than 40)
2 * 6 = 12 (less than 40)
2 * 7 = 14 (less than 40)
2 * 8 = 16 (less than 40)
Continuing this process, we find that there are no combinations that result in a product greater than or equal to 40.
Step 3: Calculate the odds against the product being greater than or equal to 40
Since there are no favorable outcomes, the number of favorable outcomes is 0. Therefore, the odds against the product being greater than or equal to 40 are 0:64, which can also be expressed as 0.
Justification:
By systematically listing all possible outcomes and checking which ones satisfy the condition, we have determined that there are no favorable outcomes. Hence, the odds against the product being greater than or equal to 40 are 0:64.