Please explain to me how to solve this problem---TRIAL QUESTIONS
1. A technician needs to inspect 5 different machines in a day. In how many different orders can the technician schedule the inspections?
2. A quality inspector selects 3 circuit boards from a batch of 7 for testing. How many different groups of 3 boards can be selected?
3. A diagnostic tool has a 95% detection rate. If disease prevalence is 2%, what concept is required to compute true detection rate in population?
4. Two pumps operate independently. The probability that Pump A fails is 0.2, and the probability that Pump B fails is 0.15. What is the probability that both pumps fail on the same day?
5. A device produces output X={0,1,3} with probabilities 0.5,0.3,0.2 respectively. Find E(X).
6. If P(Y=0)=0.3 and P(Y=1)=0.4, what must P(Y=2) be?
7. A pie chart is most effective for showing parts of a whole in categorical data.
8. The number of ways to arrange 5 books on a shelf is 5
9. If two events are independent, the probability of their intersection equals the product of their probabilities
10. The area under the probability distribution function of a continuous random variable is always 1
11. A shipment contains 4 defective and 6 non-defective microchips. If two microchips are selected at random without replacement, what is the probability that both are non-defective?
12. During a maintenance check, the probability that a machine overheats is 0.4, and the probability that the technician forgets a tool is 0.1. Assuming independence, what is the probability that both events occur?
视频信息
答案文本
视频字幕
A technician needs to inspect 5 different machines. Since the order of inspection matters, this is a permutation problem. We calculate 5 factorial, which equals 5 times 4 times 3 times 2 times 1, giving us 120 different possible orders.
A quality inspector selects 3 circuit boards from 7 for testing. Since the order of selection doesn't matter, this is a combination problem. We use the combination formula C of 7 choose 3, which equals 7 factorial divided by 3 factorial times 4 factorial. This simplifies to 7 times 6 times 5 divided by 3 times 2 times 1, giving us 35 different groups.
A diagnostic tool has 95% detection rate with 2% disease prevalence. To compute the true detection rate in the population, we need conditional probability, specifically Bayes' Theorem. This helps us find the positive predictive value - the probability that someone actually has the disease given they tested positive. Bayes' Theorem relates the test sensitivity, disease prevalence, and overall test positivity rate.
Two pumps operate independently. Pump A has a 0.2 probability of failure, and Pump B has a 0.15 probability of failure. Since the events are independent, we use the multiplication rule. The probability that both pumps fail equals the product of their individual failure probabilities: 0.2 times 0.15 equals 0.03 or 3 percent.
A device produces outputs 0, 1, and 3 with probabilities 0.5, 0.3, and 0.2 respectively. To find the expected value, we multiply each outcome by its probability and sum the results. E of X equals 0 times 0.5 plus 1 times 0.3 plus 3 times 0.2, which equals 1.1. This completes our overview of key probability concepts including permutations, combinations, conditional probability, independent events, and expected value.