The scores on a finite mathematics test were normally distributed with a mean of 77 and a standard deviation of 8. [a] The minimum score a student could obtain in order to receive an A is 90. What is the probability a randomly selected student receives an A? [b] The minimum score a student could obtain in order to pass the test is 60. What is the probability a randomly selected student failed the test? NEGATIVE z Scores POSITIVE z Scores Standard Normal (z) Distribution: Cumulative Area from the LEFT Standard Normal (z) Distribution: Cumulative Area from the LEFT .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .00 .01 .02 .03 .04 .05 .06 07 08 .00 -3.5 use 0.0001 for areas below -3.49 0.0 .5000 5040 .5080 .5120 5160 5199 5239 .5270 5319 .5359 and lower -3.4 0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0002 0.1 .5398 5438 5478 .5517 5557 5596 5636 5675 .5714 .5753 0.2 .5793 5832 .5871 .5010 5948 5987 6026 .6084 6103 .8141 -3.3 .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004 .0003 0.3 8179 6217 .8255 .8293 6331 6368 6406 .8443 .8480 R490 .8517 -3.2 .0007 .0007 .0008 .0006 .0008 .0006 .0008 .0005 .0005 .0005 0.4 .6554 .6591 .8828 .8884 6700 6738 .6772 .6808 .6844 .8879 -3.1 .0010 .0009 .0009 .0009 .0008 .0008 .0008 .0008 .0007 .0007 0.5 .8915 .6950 .8985 .7019 7054 7088 .7123 .7157 .7190 .7224 -3.0 .0013 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010 .0010 0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 -2.9 .0019 .0018 .0018 .0017 .0016 .0016 .0015 .0015 .0014 .0014 0.7 .7580 7811 .7842 .7673 .7704 7734 .7764 .7794 .7823 .7852 -2.8 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020 .0019 0.8 .7881 .7910 .7939 .7967 .7995 8023 8051 .8078 8106 .8133 -2.7 .0035 .0034 .0033 .0032 .0031 .0030 .0020 .0028 .0027 .0026 0.9 .8159 8188 .8212 .8238 8264 8289 8315 .8340 .8365 .8389 -2.6 .0047 .0045 .0044 .0043 .0041 .0040 .0030 .0038 .0037 .0036 1.0 .8413 8438 .8461 .8485 8508 .8531 8554 .8577 8599 .8821 -2.5 .0062 .0060 .0050 .0057 .0055 .0054 .0052 .0051 .0049 .0048 1.1 .8843 8885 .8886 8708 8729 8749 8770 .8790 8810 8830 -2.4 .0082 .0080 .0078 .0075 .0073 .0071 .0060 .0068 .0068 .0064 1.2 .8849 8809 .8888 .8907 8925 .8944 8962 .8980 .8997 9015 czeo 9009 ZO8O- -2.3 .0107 .0104 .0102 .0000 .0096 .0004 .0001 .0089 .0087 .0084 1.3 .9032 9049 .9006 .9082 9115 9131 .9147 9182 9177 -2.2 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0116 .0113 .0110 1.4 .9192 9207 9222 9251 9265 9279 .9292 9306 .9319 9230 -2.1 .0179 .0174 .0170 .0166 .0162 .0158 .0154 .0150 .0146 .0143 1.5 .9332 9345 .9357 .9370 9382 9394 9406 .9429 .9441 9418 -2.0 .0228 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188 .0183 1.6 9452 9483 9474 9484 9495 9505 9515 .9525 9535 .9545 -1.9 .0287 .0281 .0274 .0268 .0262 .0256 .0250 .0244 .0230 .0233 17 9554 .9584 .9573 .9582 9501 9509 9608 .9616 O818 .9625 .9833 -1.8 .0350 .0351 .0344 .0336 .0329 .0322 .0314 .0307 .0301 .0204 1.8 9641 9840 .9856 .9664 9671 .9678 9688 .9003 9699 .9706 -1.7 .0446 .0436 .0427 .0418 .0400 .0401 .0392 .0384 .0375 .0367 1.9 .9713 9719 .9726 .9732 9738 .9744 9750 .9756 9761 9767 -1.6 .0548 .0537 .0526 .0516 .0505 .0495 .0485 .0475 .0465 .0455 2.0 .9772 9778 .9783 .9788 9793 9798 9803 .9808 9812 9817 -1.5 .0868 .0855 .0643 .0830 .0618 .0806 .0504 .0582 .0571 .0550 2.1 .9821 9826 .9830 .9834 9838 .9842 9848 .9850 9854 9857 -1.4 .0808 .0793 .0778 .0764 .0749 .0735 .0721 .0708 .0604 .0881 22 .9861 .9884 .9888 .9871 9875 .9878 9881 .9884 .9887 .9890 -1.3 .0968 .0951 .0934 .0918 .0901 .0885 .0880 .0853 .0838 .0823 2.3 .9893 .9896 .9898 .9901 9904 9906 9909 2011 .9913 .9916 -1.2 .1151 .1131 .1112 .1003 .1075 1056 .1038 .1020 .1003 .0985 2.4 .9918 9920 .9922 .9925 9927 9929 9931 .9932 .9934 .9936 2.5 9938 .9940 .9941 .9943 0043 9945 9946 .9948 .9049 9951 .9952 -1.1 .1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .1170 2.6 .9953 .9955 .9956 .9957 9959 9960 9961 .9062 .9963 .9964 -1.0 .1587 .1562 .1530 .1515 .1492 .1489 .1446 .1423 .1401 .1379 2.7 .9965 .9988 .9967 9969 0087 .9908 9970 9971 .9072 .9973 .9974 -0.9 .1841 .1814 .1788 .1762 .1736 1711 .1685 .1660 .1835 .1811 2.8 .9974 .9975 .9976 .9977 9977 9978 9979 .9079 .9980 .9981 -0.8 2119 .2090 2061 2033 .2005 .1977 .1949 .1922 .1894 .1867 2.9 .9981 .9982 .9982 .9983 9984 9084 9985 .9085 9986 .9986 -0.7 2420 2389 2358 2327 2206 2266 2236 2206 .2177 2148 3.0 .9987 .9987 .9987 .9988 9988 .9989 9989 .9089 .9990 .9990 -0.6 2743 2709 .2676 2643 .2611 .2578 2546 2514 2483 2451 3.1 .9990 .9991 9001 .9991 9992 .9992 9992 .9092 .90026 .9003 -0.5 3085 .3050 .3015 2981 2946 2912 2877 2843 2810 2776 3.2 .9993 .9993 .9094 .9994 9994 9904 9994 .9005 .9005 .9995 -0.4 3446 3400 .3372 3336 .3300 .3264 .3228 3192 .3156 .3121 3.3 .9995 .9995 .9005 .9996 9998 9090 9990 .9096 .9096 .9997 -0.3 .3821 3783 .3745 3707 .3660 3832 .3504 3557 3520 3483 3.4 .9997 9997 .9997 .9997 9997 .9097 9997 .9097 .9097 .9908 -0.2 4207 4168 4129 4090 .4052 4013 .3974 .3936 .3897 3850 3.5 use .9009 for areas above 3.49 -0.1 .4602 4562 .4522 4483 4443 4404 4364 4325 4286 4247 and higher -0.0 5000 4960 4920 4880 4840 4801 4761 4721 4681 4641

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Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.