 This is a video about using the Poisson probability tables to find probabilities connected with the Poisson distribution. You'll see the probability tables in your formula book and they look something like this. They have much in common with the probability tables for the binomial distribution. In particular, they give the cumulative probabilities, the probability that x is less than or equal to something. And they only have information for some values of lambda. Let's have a look at a couple of examples to see how to use them. First of all, suppose that x has the Poisson distribution where the expected number of events is 6.5. Let's find the probability that x is greater than 4 and less than or equal to 7. Well, the first thing is to get clear about exactly which numbers this means. If x is greater than 4 and less than or equal to 7, it can be 5, 6 or 7. Now that's all the numbers from 0 to 7. Take away the numbers from 0 to 4. So the probability that x is 5, 6 or 7, the probability that x is greater than 4 and less than or equal to 7, is the probability that x is less than or equal to 7. Take away the probability that x is less than or equal to 4. This step is essential because these are the only sorts of probability we can look up in the tables. The probability that x is less than or equal to something. Okay, so we look at the tables and we need to find where lambda is 6.5. Once we've found that, we need to look across from where x is equal to 7 and find the probability in the right column. It's 0.6728. Then we look across from 4 and again find the probability in the relevant column. And it's 0.2237. So now we can say that the probability that x is greater than 4 and less than or equal to 7, is 0.6728. Take away 0.2237, which turns out to be 0.4491. My other example is a practical example to do with asteroids hitting the Earth. You probably know that asteroids hit the Earth fairly frequently, although fortunately most of them aren't very big. There's evidence of much bigger asteroids, however. For example, this is a photograph of the Barringer crater in Arizona. And here's another one called Wolf Creek in Western Australia. Some of them are rather beautiful. For example, here is a photograph of Gossy's Bluff, which is about 200 kilometres west of Alice Springs. You may be wondering why the hills are sticking out of the Earth if they're supposed to be the result of an impact by an asteroid. But in the middle of an asteroid crater, you do sometimes get things sticking out. Just as when something falls into a body of water, there can be a raised part in the centre. Here is a photograph of Gossy's Bluff from the International Space Station. Now I read somewhere recently that the frequency of asteroid impacts decreases in proportion to the cube of the asteroid's diameter. With asteroids of one kilometre or more in diameter striking the Earth every 500,000 years on average. By the way, I'm using the word asteroid to cover all sorts of extraterrestrial object that might impact the Earth. So in particular, I'm not really distinguishing between asteroids and comets and meteors. Now there's a lot we could deduce from this statement, but let's tackle the following question. What's the probability that the Earth is struck by at least three asteroids of radius 10 metres or more in the next decade? Well first of all it helps to convert the figure of one asteroid every 500,000 years into a rate. And the rate will be 1 divided by 500,000 years. Which is 0.00000002 per year. So asteroids of diameter one kilometre or more strike at the rate of 0.00000002 per year. What about asteroids of diameter one metre? Well here we've decreased the diameter by a factor of a thousand. But we're told that the rate is inversely proportional to the diameter. So if we've decreased the diameter we need to increase the rate. In fact we're told that if we decrease the diameter by an amount we need to increase the rate by the cube of that amount. So we need to multiply the rate of 0.00000002 per year by a factor of a thousand cubed. And that gives us the rate 2000 per year. So now we know that asteroids of diameter one metre strike the Earth at a rate of 2000 every year. Having worked out the rate for asteroids of diameter one metre we can work out the rate for asteroids of diameter 20 metres. This is what we're interested in because an asteroid of radius 10 metres will have a diameter of 20 metres. Now to turn one metre into 20 metres we need to multiply by a factor of 20. And if we multiply that diameter by a factor of 20 we'll need to divide the rate by a factor of 20 cubed. So now we need to do the sum 2000 divided by 20 cubed which gives us the answer 0.25 per year. So this shows that asteroids of diameter 20 metres strike the Earth at a rate of 0.25 per year. Let's say that x is the number of asteroid strikes in the next decade. Then lambda will be 2.5 because if we expect 0.25 strikes per year then in 10 years we expect 2.5 strikes. So x will have the Poisson distribution with parameter two and a half. Now the question is asking for the probability that x is greater than or equal to three. And that would be one take away the probability that x is less than or equal to two. Looking at the tables we find where lambda is 2.5 and we come across from two to find the probability in the relevant column. And that turns out to be 0.5438. So the probability that we're looking for will be one take away 0.5438 which is 0.4562. So roughly speaking there's a probability of about 0.46 that the Earth will be struck by at least three asteroids of radius 10 metres or more in the next decade. In case this seems rather high it's worth pointing out that in 1908 the Earth was struck by an enormous asteroid in the middle of Russia. The exact size of it isn't known but it's thought that the energy released by the explosion was a thousand times greater than the energy released when the atomic bomb was detonated above Hiroshima. Here is a photograph taken many years later which gives you some sense of the devastation. Here is an eyewitness account of the impact. At breakfast time I was sitting by the house at Vanavara Trading Post facing north. I suddenly saw that directly to the north over on called Tunguska Road the sky split in two and fire appeared high and wide over the forest. The split in the sky grew larger and the entire northern side was covered with fire. At that moment I became so hot that I couldn't bear it as if my shirt was on fire. From the northern side where the fire was came strong heat. I wanted to tear off my shirt and throw it down but then the sky shut closed and a strong thump sounded and I was thrown a few metres. I lost my senses for a moment but then my wife ran out and led me to the house. After that such noise came as if rocks were falling or cannons were firing. The earth shook and when I was on the ground I pressed my head down fearing rocks would smash it. When the sky opened up hot wind raced between the houses like from cannons which left traces in the ground like pathways and it damaged some crops. Later we saw that many windows were shattered and in the barn a part of the iron lock snapped. Let us end here on this sober note. Thank you for watching.