Probability Distributions

The probability that a randomly selected value from a standard normal distribution will fall between -1 and 1 is given by the area under the normal curve between these two values. This area can be calculated using the standard normal distribution table or a calculator and is approximately 0.6826.

To find the probability that X will be less than -2, we look up the probability that a standard normal random variable will be less than -2 in the standard normal distribution table or use a calculator. This probability is approximately 0.0228.

To find the probability that a randomly selected value from this distribution will be between 80 and 120, we first standardize the values using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z1 = (80 - 100) / 15 = -1.33 and Z2 = (120 - 100) / 15 = 1.33. Then, we look up the probability that a standard normal random variable will be between -1.33 and 1.33 in the standard normal distribution table or use a calculator. This probability is approximately 0.8413.

To find the probability that X will be greater than 1, we look up the probability that a standard normal random variable will be greater than 1 in the standard normal distribution table or use a calculator. This probability is approximately 0.1587.

To find the probability that a randomly selected value from this distribution will be less than 80, we first standardize the value using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z = (80 - 100) / 15 = -1.33. Then, we look up the probability that a standard normal random variable will be less than -1.33 in the standard normal distribution table or use a calculator. This probability is approximately 0.0228.

To find the probability that X will be between -1 and 1, we look up the probability that a standard normal random variable will be between -1 and 1 in the standard normal distribution table or use a calculator. This probability is approximately 0.6826.

To find the probability that a randomly selected value from this distribution will be between 70 and 130, we first standardize the values using the formula Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, Z1 = (70 - 100) / 15 = -2 and Z2 = (130 - 100) / 15 = 2. Then, we look up the probability that a standard normal random variable will be between -2 and 2 in the standard normal distribution table or use a calculator. This probability is approximately 0.8413.

Nilakantha Somayaji developed a new method for calculating the probability of an event. His method was more accurate than the methods that were used by previous mathematicians.

The sum of probabilities for all possible values of X must be equal to 1. So, we have: P(X = 1) + P(X = 2) + P(X = 3) = k * (1^2) + k * (2^2) + k * (3^2) = k * (1 + 4 + 9) = k * 14 = 1. Therefore, k = 1/14.

Since A and B are independent, the probability of both occurring is the product of their individual probabilities. So, P(A and B) = P(A) * P(B) = 0.4 * 0.6 = 0.24.

The probability of selecting a defective item on the first draw is 10/100. After the first draw, there are 9 defective items and 89 non-defective items remaining. The probability of selecting a defective item on the second draw is 9/99. Therefore, the probability of selecting two defective items is (10/100) * (9/99) = 1/100.

The probability of preferring coffee or tea is the sum of the probabilities of preferring coffee and preferring tea, minus the probability of preferring both. So, P(coffee or tea) = P(coffee) + P(tea) - P(coffee and tea) = 0.6 + 0.3 - 0.1 = 0.9.

The probability that X is less than or equal to 0.5 is given by the integral of f(x) from 0 to 0.5: P(X ≤ 0.5) = ∫[0, 0.5] 2x dx = [x^2]_[0, 0.5] = 0.5^2 - 0^2 = 0.25.

We can use the standard normal distribution (Z) to calculate this probability: P(X > 100) = P((X - 95)/10 > (100 - 95)/10) = P(Z > 0.5) = 1 - P(Z ≤ 0.5) = 1 - 0.6915 = 0.1587.

The probability of preferring only coffee is the probability of preferring coffee minus the probability of preferring both coffee and tea. So, P(coffee only) = P(coffee) - P(coffee and tea) = 0.4 - 0.2 = 0.2.