Two questions:
1. Assume a building has two independent security alarm systems. The first system works for 80% of the time and the second works 70% of the time. What percentage of time is the building protected by at least one security alarm system? Please explain.
2. There are 100 ping pong ball in a bag. some of them are white and the rest are yellow. Latoya is asked to randomly pick out 2 ping pong balls (1 at a time with replacement). She finds out that the chance for her to get 2 white balls is exactly 49%. How many white balls are in the bag?
Calculate the probability that the building is protected by NO alarm. Subtract from 1.0 gives you the probability required.
P(≥1)=1-(1-80%)*(1-70%)
Let P(W)=x
Probability of picking two whites
P(WW)=x²=0.49
x=?
got it thank you very much
You're welcome!
1. To determine the percentage of time the building is protected by at least one security alarm system, we need to calculate the complement of the probability that both systems fail simultaneously.
Let's start with system 1, which works 80% of the time. The probability of this system failing is 1 - 0.8 = 0.2 (20% chance).
Moving on to system 2, which works 70% of the time. The probability of this system failing is 1 - 0.7 = 0.3 (30% chance).
Since the two alarm systems are independent, we can calculate the probability of both systems failing simultaneously by multiplying the individual failure probabilities: 0.2 * 0.3 = 0.06 (6% chance).
The complement of this probability represents the percentage of time the building is protected by at least one system. Therefore, the percentage is 100% - 6% = 94%.
2. Let's use a proportional approach to solve this problem.
Since we know that the chance of picking two white balls is 49%, we can represent it as a fraction of the total number of ways of picking two balls from 100. Mathematically, this can be expressed as:
(Probability of picking two white balls) = (Number of ways to pick 2 white balls) / (Total number of ways to pick 2 balls from 100)
Let's assume there are "x" white balls in the bag. The number of ways to choose 2 white balls out of "x" would be given by the combination formula: C(x, 2) = x! / (2! * (x - 2)!), which simplifies to (x * (x-1))/2.
Now, the total number of ways to pick 2 balls from 100 is C(100, 2) = 100! / (2! * (100 - 2)!), which simplifies to 100 * 99 / 2.
Using this information, we can rewrite the equation as:
0.49 = (x * (x - 1)) / (100 * 99 / 2)
Simplifying further:
0.49 = (x * (x - 1)) / 4950
Now, we can solve this quadratic equation:
x^2 - x - (0.49 * 4950) = 0
We can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, with a = 1, b = -1, and c = -(0.49 * 4950).
Simplifying further:
x = (1 ± √(1 - 4 * (1) * (-(0.49 * 4950)))) / 2
x = (1 ± √(1 + (0.49 * 4950 * 4))) / 2
After calculating the expression inside the square root and simplifying further, we find:
x ≈ 7.74 or x ≈ 101.26
Since the number of white balls must be a whole number, we conclude that there are approximately 8 white balls in the bag.