Problem 8 [10 points = 2+2+2+2 + 2] Department of Public Safety analysts believe that the average car collision rate per day at a busy inter- section is 0.5 per day. The number of collisions during an observation period of NINE days is denoted as M. Use cumulative probabilities for Poisson distribution (Table 2) to answer questions below. 1. Determine the chance to have more than FIVE collisions, P [M > 5] 2. Find how likely is to have at least FIVE collisions, P [M 5] 3. Find probability of having fewer than FIVE collisions, P [M < 5] 4. Derive how likely it is to have exactly FIVE collisions, P[M=5] = 5. Derive the chance that there will be 4 to 8 collisions, including the endpoints, P [4 M 8] Solution Problem 7 [15 points = 3 + 4 + 4 + 4] Assume that a textbook contains n = 400 pages and the chance that there is at least one typo on a randomly selected page is = 0.015. Variable T denotes the total count of typos found in the textbook. Use cumulative probabilities for Poisson distribution (Table 2) to answer questions below. 1. Find probability that the textbook contains at least five typos P, [T 5] 2. Determine the chance that T is higher than three and below seven P [3 TABLE 2 Poisson Probabilities k Tabulated values are P(X k) = p(xi). (Values are rounded to four decimal places.) x=0 k 0.10 0.20 0.30 0.40 0.50 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 0 0.9048 0.8187 0.7408 0.6703 0.6065 0.3679 0.2231 1 0.9953 0.9825 0.9631 0.9384 0.9098 0.7358 0.5578 2 0.9998 0.9989 0.9964 0.9921 0.9856 0.9197 0.8088 3 1.0000 0.9999 0.9997 0.9992 0.9982 0.9810 0.9344 1.0000 1.0000 0.9999 0.9998 0.9963 0.9814 0.1353 0.0821 0.0498 0.4060 0.6767 0.8571 0.9473 0.8912 1.0000 1.0000 0.9994 0.9955 0.9834 0.9580 0.9161 0.9999 0.9991 0.9955 0.9858 0.9665 1.0000 0.9998 0.9989 0.9958 0.9881 8 1.0000 0.9998 0.9989 0.9962 9 10 11 12 13 14 15 16 17 18 19 20 0.0302 0.0183 0.2873 0.1991 0.1359 0.0916 0.5438 0.4232 0.3208 0.2381 0.7576 0.6472 0.5366 0.4335 0.8153 0.7254 0.6288 0.8576 0.7851 0.9347 0.8893 0.9733 0.9489 0.9901 0.9786 1.0000 0.9997 0.9989 0.9967 0.9919 0.9999 0.9997 0.9990 0.9972 1.0000 0.9999 0.9997 0.9991 1.0000 0.9999 0.9997 1.0000 0.9999 1.0000 0.0111 0.0067 0.0041 0.0025 0.0611 0.0404 0.0266 0.0174 0.1736 0.1247 0.0884 0.0620 0.3423 0.2650 0.2017 0.1512 0.5321 0.4405 0.3575 0.2851 0.7029 0.6160 0.5289 0.4457 0.8311 0.7622 0.6860 0.6063 0.9134 0.8666 0.8095 0.7440 0.9597 0.9319 0.8944 0.8472 0.9829 0.9682 0.9462 0.9161 0.9933 0.9863 0.9747 0.9574 0.9976 0.9945 0.9890 0.9799 0.9992 0.9980 0.9955 0.9912 0.9997 0.9993 0.9983 0.9964 0.9999 0.9998 0.9994 0.9986 1.0000 0.9999 0.9998 0.9995 1.0000 0.9999 0.9998 1.0000 0.9999 1.0000