1. When transmitting messages from a point A to a point B, out of every 40 messages 6 need to be corrected by applying error correcting codes. What is the probability that in a batch of 200 messages sent from A to B, there will be between 38 and 42 messages that will have to be corrected.

2. The probability of an event A occurring in each of a series of independent trials is 2/3. Find the distribution function of the number of occurrences of A in 9 trials.

3. The probability that a network will be shut down on any given day is .0003. What is the probability that the network will be shut down twice in 5000 days.

4. 82% of items are "regular" and 77% of regular items are "top quality." What is the probability that a randomly picked regular item is of the top quality?

5. You have two boxes. 27 out of 30 items in the first box are "regular" and 11 out of 15 items in the second box are "regular." A randomly chosen item is taken from the second box and placed in the first. What is the probability that a randomly picked item from the first box is regular?

1.
The number of messages that need to be corrected out of 200, X, follows a Normal distribution with mean 200(6/40) = 30 and variance 200(6/40)(1-6/40) = 25.5. We then need to find out

Pr(38-0.5 < X < 42+0.5)

Where we subtract and add 0.5 in the appropriate places for a continuity adjustment to the normal distribution. This calculation then becomes:

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