What is the probability that the total of two dice will be greater than or equal to 9 Given that the first die is a 4?

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Next section: Probability of A and B A conditional probability is the probability of an event given that another event has occurred. For example, what is the probability that the total of two dice will be greater than 8 given that the first die is a 6? This can be computed by considering only outcomes for which the first die is a 6. Then, determine the proportion of these outcomes that total more than 8. All the possible outcomes for two dice are shown below:

There are 6 outcomes for which the first die is a 6, and of these, there are four that total more than 8 (6,3; 6,4; 6,5; 6,6). The probability of a total greater than 8 given that the first die is 6 is therefore 4/6 = 2/3. More formally, this probability can be written as: p(total>8 | Die 1 = 6) = 2/3. In this equation, the expression to the left of the vertical bar represents the event and the expression to the right of the vertical bar represents the condition. Thus it would be read as "The probability that the total is greater than 8 given that Die 1 is 6 is 2/3." In more abstract form, p(A|B) is the probability of event A given that event B occurred.

Next section: Probability of A and B

Solution:

When two fair six-sided dice are rolled

We have to find the probability that the sum is 9 or higher

About 36 different combos are present for the two dice i.e. 6 possibilities for the first dice and 6 possibilities for the second.

Among the 36, 10 have the sum 9 or higher

3 and 6

4 and 6

4 and 5

5 and 4

5 and 5

5 and 6

6 and 3

6 and 4

6 and 5

6 and 6

The favorable outcomes are {(3,6),(4,6),(4,5),(5,4),(5,5),(5,6),(6,3),(6,4),(6,5),(6,6)}

Number of favorable outcomes = 10

Probability = 10/36 = 5/18

Therefore, the probability that the sum is 9 or higher is 5/18.

Summary:

If you roll two fair six-sided dice, the probability that the sum is 9 or higher is 5/18.

This is an example of a conditional probability question. You are given that at least one of the die is a $6$. That means this is not accounted into your probability.

Now consider, given one die is a $6$, the probability of summing to a $9$ or greater. The possibilities are $6+3, 6+4, 6+5, 6+6$. Now, either the first die rolls a $6$ or the second die rolls a $6$. So, each possibility is counted twice except $6+6$ because in this case both of the dice roll a $6$.

Then, there are $7$ ways, given at least one die is a $6$, to roll a $9$ or greater. In total, however, you also have $6+1, 6+2, 2+6, 1+6$. Therefore there are $11$ total possibilities.

Then, the probability is $\frac{7}{11}$. Hope this helps!

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Concept:

Conditional Probability: The probability of an event occurring given that another event has already occurred is called a conditional probability.

The probability of occurrence of any event A when another event B in relation to A has already occurred is known as conditional probability. It is depicted by P (A|B).

${\rm{P}}\left( {{\rm{A}}/{\rm{B}}} \right) = \frac{{{\rm{P}}\left( {{\rm{A}} \cap {\rm{B}}} \right)}}{{{\rm{P}}\left( {\rm{B}} \right)}}$

Calculation:

Let A = first die is {1, 2, 3, 4, 5, 6}

The probability of first die is a 5,

∴ P (A) = 1/6

Let B = total of two dice is greater than 9

Total outcomes when two dice are rolled = 6 × 6 = 36

Possible outcomes for A and B: (5, 5), (5, 6)

∴ P (A ∩ B) = 2/36 = 1/18

Applying the conditional probability formula we get,

$\text{P}\left( \frac{\text{B}}{\text{A}} \right)=\frac{\text{P}\left( \text{A}\cap \text{B} \right)}{\text{P}\left( \text{A} \right)}=\frac{\left( {}^{1}\!\!\diagup\!\!{}_{18}\; \right)}{\left( {}^{1}\!\!\diagup\!\!{}_{6}\; \right)}=\frac{1}{3}$

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Answer

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Hint: We will use the definition of the probability to solve this question. We will list all the possible outcomes and then we will use the formula of the probability as: Probability of an event = the total number of possible outcomes/ the total number of outcomes.

Complete step-by-step answer:

We are given that the total dice thrown are 2. And when the first die is rolled, we get a 5. The probability is defined as the likeliness of an event to occur. It is defined as the total number of possible outcomes divided by the total number of events.First, we will list the total outcomes when there is a 5 on the first die.So, the total outcomes will be: {(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}Now, we are given a condition that we need to find the probability that the total of two dice will be greater than 9.So, the possible outcomes which has a total greater than 9 are: {(5, 5), (5, 6)}Using the definition of probability, Probability = the total number of possible outcomes /the total number of outcomes Substituting the value, we getProbability = $\dfrac{2}{6} = \dfrac{1}{3}$

Hence, the probability of getting the total of two dice greater than 9 is $\dfrac{1}{3}$.

Note: You can also solve this question by using the method of conditional probability. It is better if you write all the outcomes and then select the required outcomes. Else, it will become a bit more confusing. And, always check that the probability obtained must lie between 0 and 1. Here as well, the probability is 0.333 which is between 0 and 1.