Three bags, each containing ten balls, with the following combinations:
1) 3 red, 7 black
2) 8 red, 2 black
3) 4 red, 6 black
One of the bags is randomly selected, and a ball is drawn. If the ball drawn is red, what is the probability that it is taken from the third bag?
Use Bayes’ equation to get the answer:
The Stern Review has been one of the most influential economic reports on climate change. It is an independent review commissioned by the chancellor of the exchequer of the UK to assess climate change and its economics.
The report acknowledges the urgency required to control climate change. According to the report, the loss due to climate change is about 5% of GDP each year. He recommends carbon taxes, about 1% of the GDP, as the way to finance mitigation strategies.
Social Cost of Carbon – The Stern Review Read More »
Here is a story of the survival of the fittest. It seems it is caused by global warming or some other confounding factor. A recent publication by Kang et al. in “Environmental Biology of Fishes” tells a curious – potentially scary – case of things to come.
The team noticed a sudden spike in a particular variety of fish off the coast of southeast China. This weirdly named fish, the Bombay Duck, has had a ten-fold population growth in the last decade. Bombay Duck (Harpadon nehereus) is fish that can survive a low Oxygen environment due to a high (about 90%) water content in its tissues.
So scientists postulate that as the water temperature rises, thanks to global warming, the dissolved oxygen levels in the water drop, and makes the lives of the indigenous fish species in danger, leaving only those species that can thrive under these conditions to multiply in numbers. So a fish that did not exist in the national statistics as an independent species until recently suddenly becomes a dominant variety.
Increase of a hypoxia-tolerant fish: Environmental Biology of Fishes
An Ocean Full of Bombay Duck Read More »
A player rolls a die and can win the number of dollars equal to the number on the die except when the die shows a 6. If a 6 is rolled, the player loses $6. If the game is to be fair, what should be the cost to play?
So, what is a fair game? A fair game is something where the expected value is zero. Let’s plug in all the numbers and the unknown (the cost) in the expected value calculations.
E = (1 + 2 + 3 + 4 + 5)/6 – 6/6 = 9/6 = 1.5
The expected value is $1.5. So the game can charge $1.5 to make it fair.
We have estimated the expected value of the Powerball to be about -$1.54, but for a base prize for a jackpot of $20 mln. The money will roll over to the next if there’s no winner. So there is a probability of increasing the prize to higher and higher.
So, how can one increase the expected value? EV is a product of two parameters, but one can not modify the ‘probability’, which is fixed as in the combination calculation we did earlier. As mentioned earlier, the jackpot amount grows in case there are no winners, implying an increase in the expected value.
It is good news and bad news. The good news is that the number of tickets sold increases as the prize gets heavier and the chance of winning (as per the binomial distribution). The bad news is the number of winners who will eventually share the jackpot also increases.
I see recommendations to choose unique numbers to reduce the chance of sharing, but that looks silly. The person, shared or not (plus taxes), is more likely to get more than the $2 she spent on the ticket as the jackpot! So why waste time selecting what she thinks is unique?
Powerball – Maximising EV Read More »
Let’s summarise everything we have done so far:
| Match | Prize | Probability % | Expected Value ($) |
| Jackpot | $20,000,000 | 0.00000034 | 0.068 |
| Five whites | $1,000,000 | 0.0000085 | 0.085 |
| Four whites + red | $50,000 | 0.0001095 | 0.055 |
| Four whites | $100 | 0.00274 | 0.0027 |
| Three whites + red | $100 | 0.006899 | 0.007 |
| Three whites | $7 | 0.1724838 | 0.012 |
| Two whites + red | $7 | 0.1425866 | 0.01 |
| One white + red | $4 | 1.087223 | 0.04 |
| Red | $4 | 2.609335 | 0.1 |
| Nothing | -$2 | 95.98 | -1.9196 |
| Total | 4.02 | -1.54 |
So, there is a 95.98% chance that you win nothing, and the expected value of the affair is -$1.54. To remind you, the expected value is the money I can make in the long run.
Not to forget, if the number of tickets sold increases, there could also be multiple people winning it, splitting the prize. Here is the probability of winning the jackpot estimated using the binomial distribution.
Here is the R code of the above plot:
xx <- seq(0, 500, 50)
yy <- dbinom(x = 0, size = xx, prob = 1/292.201338)
yy1 <- dbinom(x = 1, size = xx, prob = 1/292.201338)
yy2 <- dbinom(x = 2, size = xx, prob = 1/292.201338)
yy3 <- dbinom(x = 3, size = xx, prob = 1/292.201338)
yy4 <- 1 - pbinom(3, size = xx, prob = 1/292.201338)
par(bg = "antiquewhite1")
plot(xx,yy, ylim = c(0,1), type ="l", col = "red", xlab = "Number of tickets sold (million)", ylab="Probability")
polygon(x = c(0, xx, 500), y = c(0, yy, 0), col = "#006666")
text(x = 400, y = 0.1, "0 winner", col = "red")
lines(xx,yy+yy1, col ="blue" )
polygon(c(xx, rev(xx)), c(yy+yy1, rev(yy)),col = "#009999")
text(x = 400, y = 0.45, "1 winner", col = "red")
lines(xx,yy+yy1+yy2, col = "green" )
polygon(c(xx, rev(xx)), c(yy+yy1+yy2, rev(yy+yy1)), col = "#00cccc")
text(x = 400, y = 0.75, "2 winners", col = "red")
lines(xx,yy+yy1+yy2+yy3, col = "brown" )
polygon(c(xx, rev(xx)), c(yy+yy1+yy2+yy3, rev(yy+yy1+yy2)), col = "#00ffff")
text(x = 400, y = 0.92, "3 winners", col = "red")
lines(xx,yy+yy1+yy2+yy3+yy4, col = "black" )
polygon(c(xx, rev(xx)), c(yy+yy1+yy2+yy3+yy4, rev(yy+yy1+yy2+yy3)), col = "#99ffff")
text(x = 400, y = 0.98, ">3 winners", col = "red")Powerball – What $2 can Get Read More »
So, there is a 1/292201338 chance of winning a jackpot at the Powerball. That means if the prize is $20 million, the expected value of a ticket is $20,000,000 x 1/292201338 = $0.068. Not to forget, it includes the value to the individual and the state, which takes over half of the winning amount as taxes!
The jackpot is not the only thing you get; there are a few more things you win, as follows.
| Match | Prize |
| Five whites | $1,000,000 |
| Four whites + red | $50,000 |
| Four whites | $100 |
| Three whites + red | $100 |
| Three whites | $7 |
| Two whites + red | $7 |
| One white + red | $4 |
| Red | $4 |
Let’s look at the probabilities next.
P(5W AND NR) = P(5W)xP(NR)
P(5W) = 1 / 69C5 = 1/11238513
P(NR) = (25/26)
P(5W AND NR) = (1/11238513) x (25/26)
E.V. (5W AND NR) = $1,000,000 x 25 / (11238513 x 26) = $0.085
P(4W AND 1R) = P(4W)xP(1R)
P(4W) = How many ways of getting 4 winning numbers of 5 drawn out of all combinations of 5 from 69 = How many ways of getting 4 winning numbers of 5 drawn x How many ways of the remaining 1 is not / How many ways of getting 5 winning numbers of 5 drawn from 69 = (5C4 x 64C1) / 69C5 = 320 / 11238513
P(1R) = (1/26)
P(4W AND 1R) = (320/11238513) x (1/26)
E.V. (4W AND 1R) = $50,000 x (320/11238513) x (1/26) = $0.055
P(4W AND NR) = P(4W)xP(NR)
P(4W AND NR) = (320/11238513) x (25/26)
E.V. (4W AND NR) = $100 x (320/11238513) x (25/26) = $0.0027
P(3W AND 1R) = P(3W)xP(1R)
P(3W) = (5C3 x 64C2) / 69C5 = 20160/ 11238513
P(1R) = (1/26)
P(3W AND 1R) = (20160/11238513) x (1/26)
E.V. (3W AND 1R) = $100 x (20160/11238513) x (1/26) = $0.007
P(3W AND NR) = P(3W)xP(NR)
P(3W AND NR) = (20160/11238513) x (25/26)
E.V. (3W AND NR) = $7 x (20160/11238513) x (25/26) = $0.012
P(2W AND 1R) = P(2W)xP(1R)
P(2W) = (5C2 x 64C3) / 69C5 = 416640/ 11238513
P(1R) = (1/26)
P(2W AND 1R) = (416640/11238513) x (1/26)
E.V. (2W AND 1R) = $7 x (416640/11238513) x (1/26) = $0.01
P(1W AND 1R) = P(1W)xP(1R)
P(1W) = (5C1 x 64C4) / 69C5 =3176880/ 11238513
P(1R) = (1/26)
P(1W AND 1R) = (3176880/11238513) x (1/26)
E.V. (1W AND 1R) = $4 x (3176880/11238513) x (1/26) = $0.04
P(0W AND 1R) = P(0W)xP(1R)
P(0W) = (64C5) / 69C5 = 7624512/ 11238513
P(1R) = (1/26)
P(0W AND 1R) = (7624512/11238513) x (1/26)
E.V. (0W AND 1R) = $4 x (7624512/11238513) x (1/26) = $0.1
The Math of Powerball: Think Big
The odds you’ll win the Powerball jackpot: CNBC
Powerball Probabilities Read More »
In the Powerball drawing, the numbers are selected from two containers. Five white balls from 69 balls numbered 1 through 69 and one from 26 balls (1-26). To win the Powerball jackpot, the person must watch all six balls. What is the probability of winning the jackpot if you buy a Powerball ticket?
The probability of winning the jackpot = the probability of winning five numbers from the first pot x the probability of winning the number from the second pot. And the good news is, the order doesn’t matter. So you have one way of getting out of the so many ways of drawing five white balls. And since the order doesn’t matter, as you know, it is a combination problem.
So you have one chance out of 69C5 ways of getting five white balls AND 26 ways of getting the red ball. Or 1/(69C5 x 26) = 1/292201338. In case you forgot, the formula for combination, s balls from n possibilities,
The number of combinations of n things, taken s at a time = [n!/s!(n-s)!]
Finally, is it worth spending $2 for a ticket that offers a jackpot of $20 million? We’ll see next.
Make a guess: how much brain did Einstein use? Then, how much does an average human use? Here is a clue, Einstein used the same quantity as most of us use – 100%. But, many of us believe we don’t use all of it and can improve its utilisation through regular training.
As per a survey made by Dekker et al., 48% of teachers in the UK and 46% in the NL believed in this myth. Interestingly, people maintain this belief despite their knowledge that even a tiny brain injury can inflict significant impairment on human functions.
Myths such as these appeal to us because we know the human potential, like memory, may be enhanced and attribute providing some ‘intelligence’ is the only function of the brain. The brain does a lot more than this. And seats for all body functions – vision, hearing, smell, language, number, touch, motor functions, tear production, blinking, blood flow, coughing, and countless others – are placed all over it.
Neuromyths in education: Prevalence and predictors of misconceptions among teachers, Front. Psychol., 18 October 2012
10 Percent Brain Myth: Wired
Do People Only Use 10 Percent of Their Brains?: Scientific American
Einstein’s 10% Brain Read More »
Scope neglect is a cognitive bias in which a person fails to comprehend the difference in the magnitude of numbers. A simple example is the difference between 1 billion and 1 trillion. Most people know a trillion is different, but it is difficult to imagine it is 1000 times bigger than a billion.
Scope neglect or insensitivity occurs because we are unable to visualise large numbers. When one can’t picture large numbers, they remain in abstract form, failing to create the expected level of emotional reaction.
In one study, the participants were asked what they would contribute to saving 2000, 20000 or 200,000 birds from drowning in oil-contaminated ponds. The answers were $80, $78 and $88, respectively!