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14.25 Hot Spot. Hot Spot is a California lottery game. Players pick 1 to 10 Spots (sets of numbers, each from 1 to 80) that they want to play per draw. For example, if you select a 4 Spot, you play four numbers. The lottery draws 20 numbers, each from 1 to 80. Your prize is based on how many of the numbers you picked 27% Page 1584 of 6452 Location Co match one of those selected by the lottery. The odds of winning depend on the number of Spots you choose to play. For example, the overall odds of winning some prize in 4 Spot is approximately 0.256. You decide to play the 4 Spot game and buy 5 tickets. Let X be the number of tickets that win some prize. 6452 Location 18277 of 68468 32°F Mostly cloudy 6:10 3/14 the location of the mean on your histogram. a. Xhas a binomial distribution. What are n and p? b. What are the possible values that x can take? e, Find the probability of each value of X. Draw a probability histogram for the distribution of X. (See Figure 14.2 on page 331 for an example of a probability histogram.) d. What are the mean and standard deviation of this distribution? Mark 14.26 Roulette-Betting on Red. A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots o and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully a Page 1585 32°F Mostly cloudy

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The answers to each part are as follows -

a. n refers to the number of bought tickets. p is representative of the probability that I will win some prize after taking a single ticket.

b. The values of X will be in the range 0 to 5. It is indicative of number of won tickets. Hence, values that are possible are 0, 1, 2, 3, 4 and 5.

c. The probability of each value of x will be calculated using Binomial formula. The formula is-

P(x) = (n choose x) × [tex] {p}^{x} [/tex] × [tex] {(1 - p)}^{n - x} [/tex]

In this formula, n is the number of trials, p is probability of success and x is number of successes.

Keep the values in formula for calculation -

P(0) = (5 choose 0) * 0.256^0 * (1-0.256)^(5-0)

P(0) = 0.326

P(1) = (5 choose 1) * 0.256^1 * (1-0.256)^(5-1)

P(1) = 0.410

P(2) = (5 choose 2) * 0.256^2 * (1-0.256)^(5-2)

P(2) = 0.219

P(3) = (5 choose 3) * 0.256^3 * (1-0.256)^(5-3)

P(3) = 0.062

P(4) = (5 choose 4) * 0.256^4 * (1-0.256)^(5-4)

P(4) = 0.011

P(5) = (5 choose 5) * 0.256^5 * (1-0.256)^(5-5)

P(5) = 0.001

d. For the probability histogram stating the distribution of x, the x-axis will have x values and y-axis will have the probability of each x value. The mean of distribution will be calculated by the formula -

mean = n × p

mean = 5 × 0.256

mean = 1.28

Further, the standard deviation will be calculated by the formula -

Standard Deviation = [tex] \sqrt{n × p × (1-p)} [/tex]

Keep the values in formula -

Standard Deviation = [tex] \sqrt{5 × 0.256 × (1-0.256)} [/tex]

Standard Deviation = 0.995

Learn more about probability -

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