Suppose that for a randomly selected rock in a certain region, P(granite) =.25 and P(basalt) =.75.

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1 EE 323 Homework Set 3 page 1 Problem One method of distinguishing between granite ( and basaltic ( rocks is to examine a portion of the infrared spectrum of the sun s energy reflected from the rock surface. Let 1, 2 and 3 denote measured spectrum intensities at three different wavelengths; typically, for granite 1 < 2 < 3 whereas for basalt 1 > 2 > 3. When measurements are made remotely (using aircraft), various orderings of i s may arise whether rock is basalt or granite. Flights over regions of known compositions have yielded the following information. Granite Basalt 1 < 2 < 3 60% 10% 1 < 3 < 3 25% 20% 3 < 1 < 2 15% 70% Suppose that for a randomly selected rock in a certain region, granite).25 and basalt).75. a. Show that granite 1 < 2 < 3 ) > basalt 1 < 2 < 3 ). If measurements yielded 1 < 2 < 3, would you classify the rock as granite or basalt? b. If measurements yielded 1 < 3 < 2, how would you classify the rock? Answer the same question for 3 < 1 < 2. c. Using the classification rules indicated in parts (a) and (b), when selecting rock from this region, what is the probability of an erroneous classification? [hint: either G could be classified as B or B as G, and and are known.] Equations Conditional Probability (p. 67 of text) Given an event B has occurred, this event B usually affects the probability assigned to the event A. The notation A will represent the conditional probability of A given that the event B has occurred. For any two events A and B with > 0, the conditional probability of A given that B has occurred is defined by (Definition on p. 69 of text): A A Equation 1 Conditional Probability (p.41 & 42 of The Cartoon Guide to Statistics )

2 EE 323 Homework Set 3 page 2 We call it the conditional probability that event A will occur, given the condition that event C has already occurred. We write A C) and we say, The probability of A, Given C. We translate this into a formal definition: the conditional probability of E, given F, is E F) E F) Equation 2 F) Bayes Theorem (p. 74 of text) Let A 1,A 2,...,A n be a collection of n mutually exclusive and exhaustive events with a i ) > - for I1,...,n. then for any other event B for which > 0 AK B Ak ) Ak ) Ak ; k 1,..., n n B A ) A ) i 1 i i Equation 3 Bayes Theorem (p. 50 of, The Cartoon Guide to Statistics ) It computes A from A) and the two conditional probabilities B A) and B A ). You can derive it by noting that the big fraction can be expressed as A A + A' A A Equation 4 Solution G {rock is granite} B {rock is basalt} { 1 < 2 < 3 } { 1 < 3 < 2 } { 3 < 1 < 2 } Where 1, 2, and 3 denote measured spectrum intensities at three different wavelengths

3 EE 323 Homework Set 3 page 3 Figure 1. Tree diagram for probabilities of wavelengths of light reflected given basalt or granite. a) Show that granite 1 < 2 < 3 ) > basalt 1 < 2 < 3 ) Using equation 4 we can find the probability that the rock reflecting the light is granite given the wavelength reflected. G ) ) ) ) + (0.25)(0.60) (0.25)(0.60) + (0.75)(0.75) Using equation 4 we can find the probability that the rock reflecting the light is basalt given the wavelength reflected. ) (0.75)(0.10) P ( B ) 0.3 ) (0.225) If measurements yielded 1 < 2 < 3, would you classify the rock as granite or basalt? Now we can compare the values that we just calculated. If the reading was, I would choose granite because G ) > B ). b) If measurements yielded 1 < 3 < 2, how would you classify the rock? Again using Baye s theorem we can calculate the probability that the rock is granite given the wavelength of light reflected. 1 3

4 EE 323 Homework Set 3 page 4 G ) g) + (0.25)(0.25) (0.25)(0.25) + (0.75)(0.20) Because being basalt or granite is mutually exclusive and that these are the only two types of rock on earth we can subtract the probability that the rock is granite given the wavelength from one to find the probability that the rock is basalt. P ( B ) 1 G ) Again we compare the calculated values If I read, then I would choose basalt because B ) > G ) Answer the same question for 3 < 1 < 2. Now we can repeat the above procedure for a new wavelength. G ) P ( B ) 1 G ) 0.93 (0.25)(0.15) (0.25)(0.15) + (0.75)(0.70) If I read, I would choose basalt because B ) > G ) c) Using the classification rules indicated in parts (a) and (b), when selecting rock from this region, what is the probability of an erroneous classification? [hint: either G could be classified as B or B as G, and and are known.] This question was a little harder to understand than the previous two. For each wavelength recorded there is a possibility that the wavelength reflected by basalt in the late afternoon sun underwater with the added distraction of moonbeams could be the same wavelength that is characteristic of granite. This may be emphasized by comparing three ven diagrams.

5 EE 323 Homework Set 3 page 5 As we can see while flying over granite the probability that we will record a value where 3 < 1 < 2 is 15%. If I read this value I would classify this rock as basalt. erroneous classification) P[(selecting B when actually or (selecting G when actually ] So we sum the intersections of the rocks and the wavelength probabilities G ) + G 321 ) + B ) Now we can rearrange equation 4 to find the intersections using given probabilities (0.25)(0.25) + (0.25)(0.15) + (0.75)(0.10) erroneous classification) 17.5%