This lecture discusses techniques for calculating probabilities using combinatorics, and applies them to scenarios involving satellite collisions and drawing playing cards. Two scenarios are presented: 1) Calculating probabilities of different outcomes from 10 satellite conjunctions each with a 5% chance of collision. 2) Drawing 10 cards from a deck and calculating probabilities of different numbers of hearts. The chance of a heart on each draw is 1/4. Probabilities are calculated using combinations, multiplication of probabilities, and subtraction of complementary events.

1. Space Environment
Lecture 57 – Space Sustainability
Kessler Syndrome activity
Professor Hugh Lewis
SESA3038 Space Environment

2. Overview of lecture 57
• This lecture is an activity/worked example to introduce you to some
techniques involving combinatorics and probability
– The true application is space debris:
– E.g. collisions, explosions – anything involving probabilities
– We use standard playing cards to illustrate the methods
• One of the key takeaways is that even if events such as collisions have high
probabilities (e.g. “1-in-20”) the probability of many such events occurring
is always smaller
Space Environment – Space Sustainability

3. Scenario 1
Satellite collisions:
• Assume 10 satellite conjunctions (close approaches)
occur. If the collision probability for each conjunction is
0.05 (i.e. 1 in 20)…
• Questions:
1. What is the probability that there are no collisions?
2. What is the probability that there is at least one collision?
3. What is the probability that all 10 collisions occur?
4. What is the probability that there are exactly five
collisions?
5. What is the probability that there are at least five
collisions?
6. What is the probability that at least the first five
conjunctions result in collisions?
Space Environment – Space Sustainability

4. Scenario 2
Playing cards:
• A pack of ordinary playing cards has an equal number of
hearts, clubs, diamonds and spades
• We take 10 cards at random from the pack and turn
them over one by one
• Questions:
1. What is the probability that there are no hearts?
2. What is the probability that there is at least one heart?
3. What is the probability that all 10 cards are hearts?
4. What is the probability that there are exactly five hearts?
5. What is the probability that there are at least five hearts?
6. What is the probability that the first five cards turned over
are hearts?
Space Environment – Space Sustainability

5. Scenario 2
Playing cards:
• Whenever we turn over a card, there are only two meaningful
outcomes: the card is a heart or it is not a heart. The
probability it is a heart is 𝑝𝑝 = ¼ or 0.25
• What is the probability that there are no hearts?
• What is the probability that there is at least one heart?
• What is the probability that all 10 cards are hearts?
10
𝑃𝑃=0 = 1 − 𝑝𝑝 𝑛𝑛
= 1 − 0.25 10
= 0.0563 = 5.63%
10
𝑃𝑃≥1 = 1 − 1 − 𝑝𝑝 𝑛𝑛
= 1 − 1 − 0.25 10
= 0.9437 = 94.37%
10
𝑃𝑃=10 = 𝑝𝑝𝑛𝑛
= 0.2510
= 9.537 × 10−7
≈ 1 in 1 million
Space Environment – Space Sustainability

6. Scenario 2
Playing cards:
• What is the probability that there are exactly five hearts?
Number of unique combinations 𝐶𝐶 of 𝑛𝑛 = 10 cards where 𝑘𝑘 = 5
hearts are possible
𝑛𝑛
𝐶𝐶𝑘𝑘 =
𝑛𝑛!
𝑘𝑘! 𝑛𝑛 − 𝑘𝑘 !
=
10!
5! × 5!
=
3628800
120 × 120
= 252
𝑃𝑃=𝑘𝑘 = 𝑛𝑛
𝐶𝐶𝑘𝑘𝑝𝑝𝑘𝑘
1 − 𝑝𝑝 𝑛𝑛−𝑘𝑘
10
𝑃𝑃=5 = 10
𝐶𝐶5𝑝𝑝5
1 − 𝑝𝑝 10−5
= 252 × 0.255
× 0.755
= 0.0584
Space Environment – Space Sustainability

7. Scenario 2
Playing cards:
• What is the probability that there are at least five hearts?
Find the probability that fewer than five cards are hearts (fewer
calculations to do in this instance)
Then
10
𝑃𝑃<5 = 𝑃𝑃0 + 𝑃𝑃=1 + 𝑃𝑃=2 + 𝑃𝑃=3 + 𝑃𝑃=4 = 0.9219
10
𝑃𝑃≥5 = 1 − 𝑃𝑃<5 = 1 − 0.9219 = 0.0781 = 7.81%
Space Environment – Space Sustainability

8. Scenario 3
Playing cards:
• What is the probability that the first five cards turned over
are hearts?
5
𝑃𝑃=5 = 𝑝𝑝𝑛𝑛
= 0.255
= 9.766 × 10−4
≈ 1 in 1000
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9. Repeat for Scenario 1
You can apply exactly the same techniques to solve the
questions in scenario 1…
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