Relative and expected frequencies：练习题

A student flips a biased coin 200 times and records 130 heads. If the student flips the same coin another 500 times, what is the expected number of heads for these 500 flips?

A group of 250 students were asked about their favorite sport. 100 said Football, 75 said Basketball, and the rest said Tennis.

A larger group of 1200 students is expected to follow the same distribution. However, in the larger group, the actual number of students who prefer Tennis is 40 more than the expected number.

What is the actual relative frequency of Tennis in the larger group?

A spinner has 4 colored sections: Red, Blue, Green, and Yellow. The relative frequency of landing on Red after 80 spins is 0.15. After 200 spins, the total number of times it landed on Red is 36.

Calculate the relative frequency of landing on Red for the final 120 spins.

The probability of a laptop being returned within a year is 0.035. A store sells 4000 laptops.

If the relative frequency of returns for the first 1000 laptops was 0.042, what is the expected number of returns for the remaining 3000 laptops to maintain the overall expected relative frequency of 0.035?

A factory tests a sample of bulbs and finds that the relative frequency of a bulb being defective is 0.024. In a production run of 12,500 bulbs, how many bulbs would be expected to be non-defective?

A spinner has 4 sections: A, B, C, and D. The relative frequencies of landing on A, B, and C after 400 spins are \( 0.2 \), \( 0.35 \), and \( 0.15 \) respectively. Calculate the expected number of times the spinner will land on D if it is spun another 600 times.

A factory inspects 800 items and finds 48 of them are defective. If the factory produces 5000 items in a month, how many defective items would be expected?

The probability that a train arrives on time is 0.85. If there are 300 trains scheduled for a week, how many would you expect to arrive on time?

A factory produces light bulbs. Quality control checks 500 bulbs and finds 20 faulty.

(a) Calculate the relative frequency of a faulty bulb.

(b) If the factory produces 8000 bulbs in a week, estimate the number of faulty bulbs.

(c) The next week, they implemented a new process. Out of 300 bulbs, only 9 were found faulty. Is the new process effective? Justify your answer with calculations.

Two machines, A and B, produce components. In a batch of 2000 components from Machine A, 80 were found defective. In a batch of 1500 components from Machine B, 45 were found defective.

(a) Calculate the relative frequency of defective components for Machine A and for Machine B.

(b) The company needs to produce 10,000 components in total. If they use only Machine A, estimate the number of defective components they would expect.

(c) If the company wants to keep the total number of defective components below 3% for the 10,000 production run, and both machines continue at their current defect rates, what is the maximum number of components Machine A can produce? Assume Machine B produces the rest.