Two dice are drawn simultaneously. Probability to get a total of 5 is? A family has two children. Probability that both the children are boys, given that at least one of them is a... Two dice are drawn simultaneously. Probability to get a total of 5 is? A family has two children. Probability that both the children are boys, given that at least one of them is a boy, is? The rate of change of the area of a circle with respect to its radius at r cm = 6 is?
Understand the Problem
The question consists of three parts related to probability and geometry. The first part asks for the probability of obtaining a total of 5 when two dice are rolled. The second part asks for the probability that both children in a family are boys given that at least one is a boy. The third part asks for the rate of change of the area of a circle with respect to its radius at a specific radius.
Answer
1. $\frac{1}{9}$ 2. $\frac{1}{3}$ 3. $12\pi$
Answer for screen readers
- (c) $\frac{1}{9}$
- $\frac{1}{3}$
- (b) $12\pi$
Steps to Solve
- Calculating Probability of Total 5 with Two Dice
To find the total combinations that give us a sum of 5:
- Possible outcomes are (1,4), (2,3), (3,2), (4,1).
- Total combinations = 4.
- Total outcomes when rolling two dice = (6 \times 6 = 36).
Therefore, the probability is given by:
$$ P(\text{total of 5}) = \frac{4}{36} = \frac{1}{9} $$
- Calculating Probability of Two Boys Given At Least One Boy
The sample space for two children can be represented as:
- BB (both boys)
- BG (boy and girl)
- GB (girl and boy)
- GG (both girls)
Since we know at least one child is a boy, we eliminate GG, leaving us with:
- BB
- BG
- GB
Now, the favorable outcome is only BB. Thus, the probability is given by:
$$ P(\text{two boys | at least one boy}) = \frac{1}{3} $$
- Finding Rate of Change of Area of Circle with Respect to Radius
The area (A) of a circle is given by:
$$ A = \pi r^2 $$
To find the rate of change of area with respect to radius (r):
$$ \frac{dA}{dr} = 2\pi r $$
At (r = 6):
$$ \frac{dA}{dr} = 2\pi \times 6 = 12\pi $$
- (c) $\frac{1}{9}$
- $\frac{1}{3}$
- (b) $12\pi$
More Information
- The probability of getting a total of 5 from two dice rolls is $\frac{1}{9}$.
- When considering families with at least one boy, the likelihood that both children are boys is $\frac{1}{3}$.
- The rate of change of the area of a circle at a radius of 6 cm is $12\pi$.
Tips
- Miscounting Outcomes: Ensure all combinations of dice rolls are considered while calculating probabilities.
- Assuming Independence: In the second part, remember that knowing at least one child is a boy affects the probability.
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