Probability and Statistics

For a fair coin, the probability of getting a head or a tail is equal, hence the probability of getting a head is 1/2.

In a normal distribution, the area under the curve between the mean and one standard deviation is approximately 0.3413.

The expected value of a random variable is calculated as E(X) = ΣxP(X = x). In this case, E(X) = 1(1/3) + 2(1/3) + 3(1/3) = 2.

The binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p.

The probability of getting at least one head is the complement of the probability of getting no heads. The probability of getting no heads is (1/2) * (1/2) = 1/4. Therefore, the probability of getting at least one head is 1 - 1/4 = 3/4.

The central limit theorem states that the sample mean of a large number of independent, identically distributed random variables will be approximately normally distributed, regardless of the shape of the underlying distribution.

There are 36 possible outcomes when rolling two dice. The outcomes that sum to 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). Therefore, the probability of getting a sum of 7 is 6/36 = 1/6.

There are 4 kings in a standard deck of 52 cards. Therefore, the probability of getting a king is 4/52 = 1/13.

The probability of getting a type A blood group is equal to the frequency of type A blood in the population, which is 0.4.

The probability of getting a z-score between -1 and 1 in a standard normal distribution is approximately 0.6826.

Using the central limit theorem, the probability of getting a sample mean between 100 and 110 can be approximated by the probability of getting a z-score between (100 - 105) / (10 / √25) = -1 and (110 - 105) / (10 / √25) = 1 in a standard normal distribution. This probability is approximately 0.6826.

In hypothesis testing, the probability of rejecting a null hypothesis when it is true is typically set at 0.05, which is known as the significance level.

In hypothesis testing, the probability of failing to reject a null hypothesis when it is false is known as the Type II error. The probability of a Type II error depends on the significance level, the effect size, and the sample size.

The chi-square test is commonly used for testing the goodness of fit of a distribution, which assesses how well a sample fits a particular distribution.

The t-test is commonly used for testing the equality of means of two populations, assuming that the populations are normally distributed and have equal variances.