Statistics I · Unit 3 · 8 hrs
Introduction to Probability
Exam-focused notes for Introduction to Probability (Statistics I, STA169): what the TU syllabus asks and how it has actually been tested, with 7 solved past questions from this unit.
What this unit covers
- Concepts of probability
- Definitions of probability
- Laws of probability
- Bayes theorem
- prior and posterior probabilities
- Problems and illustrative examples related to computer Science and IT
Concepts of probability
Define independent and mutually exclusive events. Three groups of children 2 boys and 2 girls, 3 boys and 1 girl, 1 boy and 3 girls respectively. One child is selected at random from each group. Find the probability of selecting one boy and two girls. [5]
- Group 1: 2 boys, 2 girls (total 4) - Group 2: 3 boys, 1 girl (total 4) - Group 3: 1 boy, 3 girls (total 4) - One child selected at random from each group. - Required: $P(\text{1 boy and 2 girls})$ Mutually Exclusive Events: Two events cannot occur at the ...
Full solved answer →Define conditional probability. A problem of statistics is given to three students, A, B, and C whose chances of solving the problem are in a ratio of 2:3:5. Find the probability that (i) none of them solve the problem (ii) the problem will be solved. [5]
Conditional probability is the probability of an event $A$ occurring given that another event $B$ has already occurred. It is defined as: $$P(AB) = \frac{P(A \cap B)}{P(B)}, \quad P(B) 0$$ --- - Ratio of chances of solving: $A : B : C = 2 : 3 : 5$ - Sum of ...
Full solved answer →Bayes theorem
What is conditional probability? Three roads A, B and C lead away from a jail. A prisoner escaping from the jail selects a road at random. If road A is selected, the probability of escaping is 1/10. Similarly for road B it is 1/8 and for road C it is 1/5. i. What is the probability that the prisoner will succeed in escaping? ii. If the prisoner has succeeded in escaping, what is the probability that he had chosen the road A? [5]
- Roads: A, B, C selected at random, so $P(A) = P(B) = P(C) = \dfrac{1}{3}$ - $P(E \mid A) = \dfrac{1}{10}$ - $P(E \mid B) = \dfrac{1}{8}$ - $P(E \mid C) = \dfrac{1}{5}$ where $E$ = event of escaping. Conditional probability is the probability of an event o...
Full solved answer →A factory has three machines M1, M2, and M3 producing a large number of computer chips. Of the total daily production of items, 50% is produced on M1, 20% on M2, and 30% on M3. Records show that 4% of chips produced on M1 are defective, 2% produced of chips produced on M2 are defective and 2.5% of chips produced on M3 are defective. The occurrence of a defective chip is independent of all other chips. One chip is chosen at random from a day's total production. (i) Show that the probability of being defective is 0.0315 (ii) Given that it is defective, find the probability that was produced on machine M1. [5]
Machine Production Share Defective Rate ------------------------------------------ M1 $P(M1) = 0.50$ $P(DM1) = 0.04$ M2 $P(M2) = 0.20$ $P(DM2) = 0.02$ M3 $P(M3) = 0.30$ $P(DM3) = 0.025$ Let $D$ = event that a randomly chosen chip is defective. --- By the To...
Full solved answer →A manufacturing company employs three analytical plans for the design and development of a particular product. For cost reasons, all three are used at varying times. In fact, plan 1, 2, and 3 are used for 30%, 20% and 50% of the products respectively. The defect rate in different procedures is as follows: $P(D/P_1) = 0.01$, $P(D/P_2) = 0.03$, $P(D/P_3) = 0.02$, where $P(D/P_j)$ is the probability of a defective product, given plan $j$. If a random product was observed and found to be defective, which plan was most likely used and thus responsible? [5]
Plan Prior $P(Pj)$ Defect Rate $P(D/Pj)$ ------------------ $P1$ 0.30 0.01 $P2$ 0.20 0.03 $P3$ 0.50 0.02 Find: posterior $P(Pj/D)$ for each plan; identify the largest. $$P(D) = (0.30)(0.01) + (0.20)(0.03) + (0.50)(0.02)$$ $$P(D) = 0.003 + 0.006 + 0.010 = 0....
Full solved answer →Define Baye's theorem. Store A, B and C have 100, 75 and 50 employees and, respectively 70, 60 and 50 percent of these are women. Registration are equally likely among all employees regardless of sex. One employee resigns, and this is woman. What is the probability that she works in store B? [5]
Store Employees % Women Women Count ---------------------------------------- A 100 70% 70 B 75 60% 45 C 50 50% 25 Total 225 140 Resignations equally likely among all employees. Observed event: the resigning employee is a woman. Find $P(\text{Store B} \mid \...
Full solved answer →Laws of probability
Define mutually exclusive events and independent events in probability. A problem of mathematics is given to three students, A, B and C whose chances of solving the problem are in ratio 2 : 3 : 5. Find the probability that (i) all of them solve the problem (ii) none of them solve the problem (iii) the problem will be solved. [5]
- Chances (probabilities) of solving the problem for A, B, C are in ratio $2 : 3 : 5$. - Total ratio parts $= 2 + 3 + 5 = 10$. Individual probabilities: $$P(A) = \frac{2}{10} = \frac{1}{5}, \quad P(B) = \frac{3}{10}, \quad P(C) = \frac{5}{10} = \frac{1}{2}$...
Full solved answer →Make Unit 3 stick
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