Given \(Set\;A = \left\{ {2,\;3,\;4,\;5} \right\}\) and Set \(B = \left\{ {11,\;12,\;13,\;14,\;15} \right\}\), two numbers are randomly selected, one from each set. What is the probability that the sum of the two numbers equals 16?
0.20
This problem involves calculating the probability of a specific event occurring when selecting numbers from two different sets. Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes.
Understanding the Sets and Outcomes
We are given two sets:
- Set A contains the numbers: $\left\{ {2,\;3,\;4,\;5} \right\}$. The total count of numbers in Set A is 4.
- Set B contains the numbers: $\left\{ {11,\;12,\;13,\;14,\;15} \right\}$. The total count of numbers in Set B is 5.
We need to select one number randomly from Set A and one number randomly from Set B. The total number of possible pairs we can form is the product of the number of elements in each set.
Total possible outcomes = (Number of elements in Set A) $\times$ (Number of elements in Set B)
Total possible outcomes = $4 \times 5 = 20$.
Identifying Favorable Outcomes
We are looking for the probability that the sum of the two selected numbers equals 16. Let's find the pairs of numbers (one from Set A, one from Set B) that add up to 16:
- If we select 2 from Set A, we need $16 - 2 = 14$ from Set B. The number 14 is present in Set B. So, (2, 14) is a favorable outcome.
- If we select 3 from Set A, we need $16 - 3 = 13$ from Set B. The number 13 is present in Set B. So, (3, 13) is a favorable outcome.
- If we select 4 from Set A, we need $16 - 4 = 12$ from Set B. The number 12 is present in Set B. So, (4, 12) is a favorable outcome.
- If we select 5 from Set A, we need $16 - 5 = 11$ from Set B. The number 11 is present in Set B. So, (5, 11) is a favorable outcome.
The favorable outcomes are the pairs: (2, 14), (3, 13), (4, 12), and (5, 11).
The number of favorable outcomes is 4.
Calculating the Probability
The probability of an event is calculated using the formula:
$$ P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} $$
In this case, the event is that the sum of the two selected numbers equals 16.
Number of Favorable Outcomes = 4
Total Number of Possible Outcomes = 20
$$ P(\text{Sum = 16}) = \frac{4}{20} $$
Simplifying the fraction:
$$ P(\text{Sum = 16}) = \frac{1}{5} $$
Converting the fraction to a decimal:
$$ P(\text{Sum = 16}) = 0.20 $$
Important Questions from Numerical Computation
Ratio of the angles of a triangle are 1 : 2 : 6. What is the smallest angle?
It would take one machine 4 hours to complete a production order and another machine 2 hour to complete the same order. If both machines work simultaneously at their respective constant rates, the time taken to complete the same order is ________ hours.
Two design consultants, P and Q, started working from 8 AM for a client. The client budgeted a total of USD 3000 for the consultants. P stopped working when the hour hand moved by 210 degrees on the clock. Q stopped working when the hour hand moved by 240 degrees. P took two tea breaks of 15 minutes each during her shift, but took no lunch break. Q took only one lunch break for 20 minutes, but no tea breaks. The market rate for consultants is USD 200 per hour and breaks are not paid. After paying the consultants, the client shall have USD_remaining in the budget.
What is the value of \(1 + \frac{1}{4} + \frac{1}{{16}} + \frac{1}{{64}} + \frac{1}{{256}} + \ldots ?\)
A 1.5 m tall person is standing at a distance of 3 m from a lamp post. The light from the lamp at the top of the post casts her shadow. The length of the shadow is twice her height. What is the height of the lamp post in meters?
