Download Probability of astronomical events

STRAND: Chance & data
TOPIC:
Probability
Probability of
astronomical events
LITERACY TASK • Find the meanings of these key terms:
1. formula: .............................................................................................................................................................................
2. function:.............................................................................................................................................................................
3. random event: ...................................................................................................................................................................
4. irrational number: ............................................................................................................................................................
5. power: ................................................................................................................................................................................
6. astronomical:.....................................................................................................................................................................
7. probability: ........................................................................................................................................................................
8. time interval: .....................................................................................................................................................................
The French mathematician Simeon-Denis Poisson developed a formula for predicting the probability of random events. Before
we look at this formula, we need to examine a few functions on your calculator that may not be so familiar.
On your scientific calculator, look for the function ex. e is an irrational number similar to π. e ≈ 2.71828… We need to be able
to calculate the value of e to a given power. To do this we use the ex function.
To find the value of e2, press 2nd then ex then 2. Your calculator display should now read 7.389056099.
Use the ex function to find the value of:
1. e3
2. e5
3. e0.4
4. e–2
5. e–1.6
The other function you will need to be familiar with is the factorial function. The factorial function is denoted with ! or n!.
The factorial of a number is the product of the number and all positive integers less than the number.
For example: 5! = 5 × 4 × 3 × 2 × 1 = 120.
Use a calculator to check whether you can correctly use this function.
Use the factorial function on your calculator to find the value of:
6. 3!
7. 7!
8. 10!
9. 12!
10. 20!
RESEARCH TASK ONE
Research the life and times of Simeon-Denis Poisson and learn about some of his other contributions to mathematics.
 John Wiley & Sons Australia, Ltd 2001
STRAND: Chance & data
TOPIC:
Probability
INVESTIGATION: Probability of
astronomical events
Poisson discovered that if a random event occurs with a long-run average rate denoted by k, then the probability P that n events
n – kt
100 ( kt ) e
will occur during a certain time interval t is: P = ------------------------------n!
Consider the case of a supernova or exploding star. On average, supernovas occur four times in a millennium. Supernova
1987A in the Large Magellanic Cloud (not the Milky Way itself) was the most recent followed by Ophiuchus (1604) in the
Milky Way Galaxy.
Taking an average lifespan of 75 years, and using Poisson’s formula, we can estimate the probability of seeing a supernova in
our lifetime:
11. Find the value of k by dividing the number of times a supernova occurs in a millennium by 1,000.
12. The value of t is the time interval we are looking at, in this case, 1 lifetime of 75 years. What is the value of t?
13. The value of n is the number of time intervals we are looking at, in this case, 1 lifetime. What is the value of n?
14. Substitute the values of k, t and n into the formula and calculate the value of P as a decimal.
15. Calculate P as a percentage.
Using the method from parts 11 – 15, answer the following:
16. Meteors arrive at an average rate of 6 per hour for a single observer with a dark, unobstructed view. How likely is the
appearance of a meteor within the next 15 minutes?
17. The average rate for the appearance of a great comet is once every 10 years. What is the probability of the appearance of a
great comet within the next 20 years?
18. The average rate for crater-forming impacts anywhere on Earth is 6 per century:
(a) What is the probability of exactly one crater being formed in a 10-year period?
(b) What is the probability of exactly two craters being formed in a 10-year period?
(c) What is the probability of exactly three craters being formed in a 10-year period?
19. The average rate of an asteroid or comet destroying planet Earth is 0.000003 per century. What is the probability of such an
event happening in one million years?
20. In 1572, Tycho Brae observed a naked eye supernova in the constellation of Cassiopeia. 32 years later in 1604, Johannes
Keppler observed a naked eye supernova in the constellation of Ophiuchus:
(a) What is the probability of observing one supernova in a 32-year period?
(b) What is the probability of observing two supernovas in a 32-year period?
RESEARCH TASK TWO
Poisson’s formula can be applied to other areas besides astronomy. Consider a sporting situation:
1. Choose your favourite sports personality.
2. Find out how many matches he or she has played.
3. Find the number of times your player has achieved a landmark result. For example, a cricketer scoring a
century, a footballer kicking 5 goals or scoring 3 tries in a match, or a golfer winning a tournament.
4. Calculate the average rate at which this landmark result is achieved.
5. Use the formula to estimate the probability that this landmark effort will be achieved in the player’s next
match.
 John Wiley & Sons Australia, Ltd 2001