Respuesta :
Answer:
Identify the favorable outcomes for each event:Event A (sum less than 10): There are 36 possible outcomes when rolling two number cubes. Out of these, 33 outcomes have a sum less than 10.Event B (sum is a multiple of 3):There are 12 outcomes where the sum is a multiple of 3.Find the outcomes that satisfy both events:There are 12 outcomes that satisfy both event A and event B. These are: (1, 2), (1, 5), (2, 1), (2, 4), (3, 3), (3, 6), (4, 2), (4, 5), (5, 1), (5, 4), (6, 3), (6, 6)Apply the conditional probability formula:P(B|A) = P(A ∩ B) / P(A)P(A ∩ B) is the probability of both A and B occurring, which is 12/36 (12 favorable outcomes out of 36 total).P(A) is the probability of event A occurring, which is 33/36.Calculate the conditional probability:P(B|A) = (12/36) / (33/36) = 12/33 ≈ 0.3636Therefore, the conditional probability of event B (sum is a multiple of 3) given that event A (sum less than 10) has already occurred is approximately 0.3636, or about 36.36%.
Answer: 11/30
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Explanation
Here's a sum of dice chart
[tex]\begin{array}{|c|c|c|c|c|c|c|} \cline{1-7} & 1 & 2 & 3 & 4 & 5 & 6\\\cline{1-7}1 & 2 & 3 & 4 & 5 & 6 & 7\\\cline{1-7}2 & 3 & 4 & 5 & 6 & 7 & 8\\\cline{1-7}3 & 4 & 5 & 6 & 7 & 8 & 9\\\cline{1-7}4 & 5 & 6 & 7 & 8 & 9 & 10\\\cline{1-7}5 & 6 & 7 & 8 & 9 & 10 & 11\\\cline{1-7}6 & 7 & 8 & 9 & 10 & 11 & 12\\\cline{1-7}\end{array}[/tex]
For example, 1+6 = 7 in the upper right corner.
Another example: 3+2 = 5 in the third row, second column.
This table has 6 rows and 6 columns, which produces 6*6 = 36 total outcomes.
Here's the probability distribution for each sum.
[tex]\begin{array}{|c|c|} \cline{1-2}\text{x} & \text{P(x)}\\\cline{1-2}2 & 1/36\\\cline{1-2}3 & 2/36\\\cline{1-2}4 & 3/36\\\cline{1-2}5 & 4/36\\\cline{1-2}6 & 5/36\\\cline{1-2}7 & 6/36\\\cline{1-2}8 & 5/36\\\cline{1-2}9 & 4/36\\\cline{1-2}10 & 3/36\\\cline{1-2}11 & 2/36\\\cline{1-2}12 & 1/36\\\cline{1-2}\end{array}[/tex]
Example: There are 6 ways to roll a 7. Those 6 ways are
- 1+6 = 7
- 2+5 = 7
- 3+4 = 7
- 4+3 = 7
- 5+2 = 7
- 6+1 = 7
"Given event A occurs" means we know 100% that the dice roll to something less than 10, i.e. the sum is between 2 and 9.
Add up the numerators for P(10) through P(12), or use the sum of dice chart, to see that there are 3+2+1 = 6 ways to get either a sum of 10, 11, or 12.
That must mean there are 36-6 = 30 ways to get a sum 2 through 9.
Add up the numerators for the P(3), P(6) and P(9) fractions to get 2+5+4 = 11.
We have 11 ways to get a multiple of three out of 30 ways to get a sum less than ten.
This is how we arrive at the answer 11/30
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Another approach
P(B and A) = probability of getting a multiple of 3 and smaller than 10
P(B and A) = probability of getting 3 or 6 or 9
P(B and A) = P(3) + P(6) + P(9)
P(B and A) = 2/36 + 5/36 + 4/36
P(B and A) = 11/36
P(A) = probability of getting a sum smaller than 10
P(A) = 1 - P(getting 10 or 11 or 12)
P(A) = 1 - ( P(10) + P(11) + P(12) )
P(A) = 1 - P(10) - P(11) - P(12)
P(A) = 1 - 3/36 - 2/36 - 1/36
P(A) = 30/36
P(B given A) = P(B and A) ÷ P(A)
P(B given A) = (11/36) ÷ (30/36)
P(B given A) = (11/36) * (36/30)
P(B given A) = 11/30
