MATT: Welcome to the maths show with me, Matt Parker.Today there is a 100% probability that we will be talkingabout probability. Understanding probability is reallyuseful so you can calculate the likelihood of somethinghappening and you can avoid being tricked.
MATT: We鈥檙e going to have a look at probability trees, bycalculating the chances of getting an ace out of anormal deck of cards. In this deck there are 52 cards,and of those: 4 of them are aces.
MATT: So the chance of one card at random being an ace is 4out of 52. Which is 1 in 13.
MATT: Then the probability of a card not being an ace is theremaining 12 out of 13.
MATT: If the card was an ace then there鈥檚 either a 1 in 13chance the next one will also be an ace or a 12 in 13that it won鈥檛 and exactly the same thing happens downthe bottom: 1 in 13 for an ace and finally 12 in 13 fornot an ace.
MATT: We can fill in all the numbers, so the top branch isgetting an ace and getting an ace, and if you multiplythem together you get 1 over 169.
MATT: The two middle ones are the same, they鈥檙e both 1 outof 13 multiplied by 12 out of 13, just in different orders.They both give us 12 out of 169, and the bottom one:not getting an ace and not getting an ace gives us 144out of 169.
MATT: If the question was, what is the probability if you drawa card and then draw a second card, that one of themwill be an ace, well that鈥檚 the middle two options.
MATT: You either get an ace, then you don鈥檛, or you don鈥檛 getan ace and then you do.
MATT: So if we add them together, we get a 24 out of 169probability that you will get just the one ace.
MATT: Thank you so much for watching this episode of themaths show, with me, Matt Parker.
MATT: And we鈥檙e good, excellent.
DIRECTOR: Sorry Matt, we鈥檙e not quite done.
MATT: I thought we nailed that, it was a good take
DIRECTOR: Matt it was a great take, but you still need to do thisfrequency tree.
MATT: Oh not frequency trees. It鈥檚 been a long day, I thoughtprobability trees was done so well,
DIRECTOR: Alright fine, you don鈥檛 have to do it.
MATT: Oh excellent
DIRECTOR: Unless you roll unless you roll two sixes.
MATT: Two Sixes! I will take those odds. No problem.
MATT: Oh, what!
DIRECTOR: Roll cameras
MATT: Okay, so on a probability tree, that shouldn鈥檛 havehappened because there鈥檚 a 1 in 6 chance of a 6. A 5 in6 chance of not a 6. So two sixes is 1 out of 36.
MATT: That鈥檚 only about 3% so we are going to do frequencytrees.
MATT: We鈥檙e going to do a frequency tree about thesesuspicious dice, which I鈥檓 now going to roll a thousandtimes.
MATT: One鈥
MATT: And鈥 one thousand!
MATT: Hey, everybody! C鈥檓on, we鈥檝e got a maths show to dohere!
MATT: So after 1000 rolls, which I鈥檝e put on to a frequencytree, you can see 703 times, I got a 6 on the first diceand a 6 on the second dice.
MATT: 70.3% that is suspicious. That is nowhere near 3%.
MATT: If we look at the individual rolls, the very first roll 840times out of 1000.
MATT: 84% of the time the first dice rolled a 6. But all 3 shouldbe the same because it鈥檚 all just one dice being rolled.
MATT: The other ones, we have 111 out of 160.
MATT: That鈥檚 69.4%
MATT: and the top one there 703 out of 840,
MATT: that鈥檚 83.7 %. Of course they鈥檙e not all going to be thesame,
MATT: but the more rolls, probably the more accurate, so Ithink about 84% seems right. That is very high for theprobability of getting a 6 on a single dice
MATT: Daisy, these dice aren鈥檛 fair!
DIRECTOR: Of course they鈥檙e not, they鈥檙e joke dice. They have fivesixes and a smiley face.
MATT: Oh yeah, that鈥檚 really mean鈥 can I borrow them?
Video summary
Mathematician and comedian Matt Parker demonstrates how to handle probabilities using probability and frequency trees for students struggling with their maths GCSE.
Designed to revise and reinforce independent probability calculations, Matt tackles GCSE Foundation approaches to theoretical and experimental probability.
With clear graphics and using probability trees and successive dice rolls, we explore theoretical results and apply proportional reasoning to frequency trees which display experimental results.
Aimed at the foundation syllabus, this video does not explore dependent probabilities.
This short film is from the 成人快手 series, The Maths Show.
Teacher Notes
During the video:
- Pause while Matt is calculating his playing card probabilities to see if your students can get the correct answer.
- Elicit if students can fill in the rest of the probability tree - emphasize that the top set of branches in the second roll should be the same as the bottom set as it鈥檚 the same experiment.
- During the frequency tree experiment explore whether students fully understand where the numbers come from.
After the video:
- Explore filling in other probability trees - if students know one probability can they fill in the rest? Identify errors and key rules to ensure students don鈥檛 make the same error - 鈥榩airs鈥 of branches should always add to one - for Foundation GCSE the top and bottom set should be the same per event.
- Use and/or rules applied to probability trees and probabilities written in a table. Can students find simple 鈥榓nd鈥 probabilities from word questions - the probability of rolling a six and flipping a head?
- Extend to exploring relative frequency and theoretical probability, and proportional reasoning questions drawn from frequency trees - as a new topic this is often an under explored area.
- If time allows a simple experiment could be carried out to work out expectation vs theoretical probability, for example drawing balls or cards out of an envelope then replacing to estimate the fraction of different colour balls/cards.
Suitable for teaching maths at GCSE in England, Wales and Northern Ireland and National 4/5 or Higher in Scotland.
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