MSV 3: Most Likely Value

Dec is about to take n penalties; he knows what n is. The probability he is successful with each penalty is p (where p is a constant) independently of the other attempts, and he knows what p is.

Dec calls ‘the number of successful penalties he is about to achieve’ X, and he wants to know what the most likely value for X is. His friend Ant offers him some statistical advice.

‘Dec, Rule One is this: your most likely value for X must be a whole number, but the expectation of X need not be a whole number.’ ‘Suppose,’ says Ant, ‘you have a biased six-sided dice, where the probability of getting a 6 is five times the probability of getting each of the other numbers.’ ‘But Ant, surely the number you ‘expect’ will be a whole number?

‘It’s obvious that the most likely value we get when we roll a dice here is 6. What is the expectation here?’ ‘Well, 10a = 1, so a = 0.1, and E(X) = 45a = 4.5.’ ‘So the most likely value is a whole number, but the expectation is not.’ ‘But you can’t actually roll 4.5!’ says Dec. ‘The expectation is “an average value”,’ says Ant, ‘so you don’t need to be able to!’

‘You’ve convinced me,’ says Dec. So what is Rule Two?’ ‘Ah, now Rule Two,’ Ant says, ‘tells you that if the expectation of X IS a whole number, then that will be the most likely value.’

‘That seems reasonable,’ says Dec. ‘What happens if the expectation of X is not a whole number?’ ‘Then you need Rule Three,’ says Ant. ‘The most likely value of X is one of the whole numbers on either side of the expectation of X.’

‘Just work out the probabilities for the whole numbers on either side of the expectation of X, and pick the one that gives you the larger value. This will be the most likely value.’

‘But do you need to do that?’ asks Dec. ‘Surely you can just round the expectation to the nearest whole number! If the expectation is 8.4, then the most likely value will be 8, while if it is 8.6, the most likely value will be 9.’ ‘You would have thought that would be the case,’ said Ant with a smile. ‘But in fact there are times when the expectation is over 8.5 and the most likely value is 8! Believe me! You have to check either side.’

Good news: Ant is offering reliable advice! How do we know this? Let’s now evaluate Rules One, Two, and Three.

Answers Maybe the best thing to do to start with is to play around with an applet that shows the Binomial probabilities for various n and p. A helpful address is the following page at the excellent Waldomaths site. PoissBin1NL.jsp Waldomaths link

After a little experimentation, it seems that Rule Two is likely to be true. Can we prove it? What about Rule Three? It would seem likely that we find the most likely value by rounding the expectation of X to the nearest whole number. This, however, fails to stand up. It is not too hard to find a counter-example. Rule One is definitely true.

So rounding would suggest that the most likely value for X is 9. But P(X = 8) = … and P(X = 9) = … So the most likely value for X here is actually 8. For example, X ~ B(20,32/75) gives E(X) =

If np is a whole number, then as q < 1, then by *, np is the most likely value, so Rule Two is true. The inequality * also tells us that Rule Three is true.

is written by Jonny Griffiths Wiht thanks to Ant and Dec.