Chapter 8: Interplanetary Bodies

Science allows us to predict some events with virtual certainty. For example, the Sun will certainly rise tomorrow. If you let go of a brick, it will certainly fall to the ground. The winter will certainly be colder than the summer. But some events in nature do not occur with any kind of certainty. We use the concept of probability to describe these types of situations in science.

Coin toss

A probability is a pure number with no units. Its maximum value is one — the event definitely occurs or always occurs. Its minimum value is zero — the event does not occur or never occurs. You might hear talk about odds of 10 to 1 against a team winning a game, or a 10% chance of rain. These are both ways of describing a probability of 0.1.

If there is more than one possible outcome, each outcome can be assigned a probability. A coin can land heads or tails. In terms of probability, we would say p_heads = p_tails = 0.5 (in other words, there's a 50% probability of either outcome) and p_heads + p_tails = 1 (or, there is a 100% probability that the coin will land on either heads or tails). We are saying that we know all the possible outcomes, and that the coin is exceedingly unlikely to land on its edge! If we were rolling a six-sided die, we would describe the possible outcomes as p₁ = p₂ = p₃ = p₄ = p₅ = p₆ = 1/6 = 0.167. We also know that p₁ + p₂ + p₃ + p₄ + p₅ + p₆ = 1.

The examples of coin tossing and dice rolling are situations where the selection of outcomes is random. Remember that the use of probability does not imply a random process. For example, a professional baseball player might have a batting average of 0.330, which means any particular time he comes to bat he has a 1 in 3 chance of getting a hit. But if you stood at the plate and swung randomly at 90 mph fastballs, your probability of getting a hit would be much lower. In general, if there are n equally probable outcomes (like in the case of the dice), the probability of each outcome is:

p_n = 1 / n 

The rule of addition for probabilities says:

p_{A or B} = p_A + p_B

This rule applies if the probabilities are mutually exclusive, or disjoint. In other words, a coin can't land on both heads and tails, and a die has to land on only one of its six sides. So the probability of it landing on any particular side is 1/6. The sum of all the possible outcomes is 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1. For independent events or independent properties, we combine probabilities using the multiplication rule:

p_{A and B} = p_A × p_B

Lecture hall of about 200 people

For example, suppose an astronomy class has 200 students. Suppose also that half the people in the class are men and half are women, and that a quarter of the people in the class are seniors. If the class sits down randomly, the probability that the person next to you is a woman is 100 / 200 = 0.5. The probability that the person next to you is a senior is 50 / 200 = 0.25. Using the multiplication rule, the probability that the person next to you is a female senior is 0.5 × 0.25 = 0.125. Going back to an earlier example, the probability of tossing three heads in a row (or three tails) is ½ × ½ × ½ =0.125. Similarly, the probability of rolling three sixes in a row is (0.167)³ = 0.0047 or about 0.5%. You would not expect to see this happen, on average, until you had rolled the dice about 200 times. Random events are independent. The probability of one event does not depend on the history of previous events. So the probability of a coin coming up heads is 0.5, even if it came up heads the last 10 times in a row. The probability can only be defined as an average over a large number of events. We can't assign a probability to the decay time of a single radioactive atom, but we can know the average probability of decay for a large number of atoms.

Now we can relate this discussion to the probability of impacts from interplanetary debris. A catastrophic impact occurs about once every 100 million years. The last one occurred 65 million years ago. Does that mean another catastrophe will not occur for 35 million years? Not necessarily! Large impacts appear to be random, not periodic. If catastrophes occur on average every 100 million years, the next one might occur in 35 million years, or 150 million years, or tomorrow.

This all sounds very abstract. What is the personal danger? If p_impact = 1 over 100 million years, then over a human lifetime (in round numbers), p_impact = 100 / 10⁸ = 10^-6. If 1 in 10 people on the planet dies in such a disaster, your chance of dying is 10^-7, or one in ten million. To put this tiny number in perspective, it's somewhat less than your probability of dying by botulism. It's a thousand times less likely than your probability of dying in a plane crash, and it's 100,000 times less likely than your probability of dying in a car crash. Asteroid impacts may be good movie fodder, but they do not represent a very likely hazard in your lifetime.