Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 60$\%$ have an emergency locator, whereas 90$\%$ of the aircraft not discovered do not have such a locator. Suppose a light aircraft has disappeared.
(a) If it has an emergency locator, what is the probability that it will not be discovered?
(b) If it does not have an emergency locator, what is the probability that it will be discovered?
(a) .067 $\\$ (b) .509
all right, were given some statistics about light aircraft that disappear over a certain country. So we're given that 70% of said aircraft are discovered. So let's make a probability tree. Just d'oh, Help us visualize this a little bit better. In fact, I'm gonna take a little longer. We know that 70% of them are discovered or 0.7. We know that they're 4 30% are not discovered. In addition, we know that 60% of the point aircraft that are discovered having emergency locator. So we're just gonna say loc for locator that 60% here, therefore without a locator 0.40 here. You know, I just speaking to say I'm gonna throw zeros at the end of everything. In addition, 90% of the aircraft not discovered do not have a locator. So no locator 90% which means there's 10% of those that do have a locator. Now, in order to find the tail end of these branches of this probability tree, which is gonna multiply as we go down the line, for example, for this spot here, she's gonna be a 70% chance of being discovered and then, given that it is, discover a 60% chance of this locator 600.7 times 0.6 is point for two, and we'll go down the line. So point 7/10 0.0.4 to 8. 3 10.1 Uh, 0.10 0.3 times 0.9. All right, look, let's look at the problem for a As for the probability that airplane is not discovered given that has an emergency locator using our definition of conditional probability, we know this is the probability not having not being discovered. Intersection having a locator all over the probability of having a locator. All right, the probability of not being discovered with a locator we can just see here it's 0.3 The probability of having a locator. Well, that's just gonna be the probability of having a locator and being discovered. What's the probability of having a locator and not being discussed? It's gonna be point for two plus 20.3 which is gonna be 0.45 if you divide this out. Not is 0.67 As for part B, we're on the probability that plane will be discovered given that it does not have a locator. All right. Same general set up. We want to find the probability that it's discovered. Intersection No locator over a probability I'm not having a locator. So the probability of being discovered? No locator 0.28 The probability of not having a locator is just gonna be one minus the probability of having a locator. So that's gonna be one minus 10.45 just 0.55 If you divide this out, this is 0.509 and there you go.