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In Exercises 1–8, use Bayes’ theorem or a tree diagram to calculate the indicated probability. Round all answers to four decimal places. [HinT: See Quick Example 1 and Example 3.]
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- Using Normal Approximation. In Exercises 5–8, do the following: If the requirements of np ≥ 5 and nq are both satisfied, estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution; if np ≤ 5 or nq < 5, then state that the normal approximation should not be used. Births of Boys with n= = 8 births and p = 0.512 for a boy, find p (exactly 5 boys).arrow_forwardMany professional baseball teams (including the Cincinnati Reds and the Boston Red Sox) use Bill James's formula to estimate their probability of winning a league pennant: (Runs scored)2 (Runs scored)2 + (Runs allowed)2 Probability) of winning This formula, whose form is reminiscent of the Pythagorean theorem, is considered more accurate than just the proportion of games won because it takes into consideration the scores of the games.t Find this probability for a team that has scored 500 runs and allowed 400 runs. (Round your answer to the nearest whole percent.)arrow_forwardPart 2 of 2 What is the probability that a randomly chosen senior will have a GPA greater than 4.1? The probability that a randomly chosen senior will have a GPA greater than 4.1 is X Sarrow_forward
- You are playing a game at a carnival. You can win $5 with probability 0.15 on each round, but if you lose, you pay $2. Assume each round that you play is independent of the others. Let X be the number of rounds it takes for you to get your first win (including the first win). X is distributed [ Select ] The expectation of X is [Select ] If you stop playing after your first win, your expected winnings (i.e. net profit, or the number of dollars you win or lose from playing the game; positive if you win more money than you lose) is [ Select ] You are playing a game at a carnival. You can win $5 with probability 0.15 on each round, but if you lose, you pay $2. Assume each round that you play is independent of the others. Let X be the number of rounds it takes for you to get your first win (including the first win). X is distributed [ Select ] [ Select ] The expectation of X Geometric(p = 0.15) Exponential(\lambda = 0.15) If you stop playing aft number of dollars you winnings (i.e. net…arrow_forwardCalculate the relative frequency P(E) using the given information. Six hundred adults are polled, and 480 of them support universal health-care coverage. E is the event that an adult supports universal health-care coverage. HINT [See Example 1.] P(E) = =arrow_forwardTask 2. Provide a proof of Theorem 3.3 part a. 106 Probability and Statistics with R Theorem 3.3 If a and b are real-valued constants, then (1) Mx+a(t) = E [e*+a)*] = eat . Mx(t).arrow_forward
- Events A and B are independent with P(AB) = 0.2 and P(A'B) = 0.6. 3a. Find P(B).arrow_forwardWhat do the functions Y. YY look like? Are there any maximum or 1+1 minimum probability at any particular angle(s) for these functions? [You don't have to prove this mathematically if you can provide your reasoning clearly.]arrow_forward%A. l. https://docs.google.com/fo YO 4.4 O 3.96 O 1.8 نقطة واحدة Let X denote the number of colleges where you will apply after your results and P(X =x) denotes your probability of getting admission in x number of colleges. It is given that if x = 0 or 1 if x = 2 if x 3 or 4 otherwise kx, 2kx, P(X = x) 3= %3D k(5 - x), 0, Where k is a positive constant. Then the probability that you will get admission in at most 2 colleges is 0.625arrow_forward
- (Sec. 3.2) A student is required to enroll in one, two, three, four, five, six on the desired courseload) at a local university. Let Y the number of classes the next student enrolls themselves in. The probability that y classes are selected is known to be proportional to y+1, in other words the pmf of Y is given by p(y) = k(y+1) for y 1,...,7, and 0 otherwise (a) What is the value of k? or seven classes (depending (b) What is the probability that at most four classes are enrolled in? (c) What is the probability that a student enrolls in between three and five classes (inclusive)? y? /40 for y 1,.,7 be the pmf of Y? Explain why why not (d) Could p(y) orarrow_forwardSection 1 4. The continuous random variable X has the following probability density function S{(1+æ), 0< x < 2; 0, fx (x) otherwise. The median, m, of a continuous random variable Y satisfies Pr(Y < m) = 0.5. Find the median of X. (Choose the option closest to the answer.) 0.8 1.4 1.2 1.6 1.0 Save For Later Nextarrow_forward2. 3. 4 6. 4 6. 16 25 36 P(X=x) 90 90 90 90 90 Reduce the fraction if you can! Find P(x-D4): Find P(x3D3): Find the probability of at least 5: 5.arrow_forward
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