4. 1.Definition and example of a random variable. 2. What is the probability of dropping 6 points when throwing a die? 3. How many of 3 flowers can be made from 2 roses, 3 lilies and 3 dahlias? 4. There are 6 balls in an urn: 3 red, 2 blue and 1 white. What is the probability to remove the blue ball after the red. 5. Coin is thrown 5 times.What is the probability of 3 heads? 6. Present the distribution law X for a series of tests in the form of a table when n=3;p(A)=0.2 7 . Calculate statistic characteristics (mean,median,mode . standard deviation) for data: 10,9,9,11,12

1. A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. For example, the roll of a die can result in the random variable taking on values from 1 to 6.<br />2. The probability of rolling a 6 on a die is 1/ as there is only one way to roll a 6 out of the six possible outcomes.<br />3. The number of ways to make 3 flowers from 2 roses, 3 lilies, and 3 dahlias is 6. The possible combinations are: 2 roses, 1 lily; 1 rose, 2 lilies; 3 lilies; 2 roses, 1 dahlia; 1 rose, 2 dahlias; 3 dahlias.<br />4. The probability of removing a blue ball after removing a red ball is 2/5. There are 2 blue balls left out of a total of 5 balls remaining in the urn.<br />5. The probability of getting 3 heads when a coin is thrown 5 times is 0.3125. This can be calculated using the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success.<br />6. The distribution law X for a series of tests with n=3 and p(A)=0.2 is as follows:<br /><br />| X | 0 | 1 | 2 |<br />|---|---|---|---|<br />| P(X) | 0.512 | 0.384 | 0.064 |<br /><br />7. The statistical characteristics for the data 10, 9, 9, 11, 12 are:<br />- Mean: 10.2<br />- Median: 10<br />- Mode: 9<br />- Standard deviation: 1.118