Deal Or No Deal
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This website is the creation of Rico, a mathematician analysing the popular TV gameshow Deal Or No Deal
Do you watch Deal Or No Deal in the USA, the UK, or elsewhere? I am a UK viewer of Deal Or No Deal. The UK version is produced by Endemol for Channel 4, and hosted by the former BBC Radio 1 DJ Noel Edmonds. I've seen people make deals at the perfect time and walk away with thousands of pounds, and also disastrous deals at the wrong time, even not dealing at any stage but opening up their final box, and going away with meagre amounts such as a 1p or 10p. I believe that the best deals tend to come later in the game providing the contestant can hold their nerve, but it is possible to get too ambitious, and gamble foolishly by dealing too late, just as it is possible to be excessively cautious and deal too early. Consequently I have created some scenarios of boxes left at various stages of the game (suitcases in the US version), and an analysis of whether or not it is sensible to deal based upon the statistical odds, the probabilities.
Sometimes, the banker offers high amounts of money to deal relatively early in the game. Usually most offers of anything upwards of £20,000 any time up to and including the 11 box stage are worth taking statistically, though contestants are encouraged not to deal too early because the general perception is that this could result in a less entertaining game. So on that basis we will focus from the 11 box stage onwards.
Interestingly, in an analysis of the first 362 shows (excluding Show 312 not aired) of the UK version of Deal Or No Deal, the banker's offer increased from the 8 box stage to the 5 box stage exactly 228 times, 63.15% (hypothetical post deal offers included), and from the 5 box stage to the last two boxes, 149 times 41.3% (hypothetical post deal offers included). This suggests that a bit of courage helps most of the time, but at the same time there is danger in getting too bold. Let's consider whether the probabilities make sense of this analysis.
First, let's assume that we have 11 boxes left of the original 22. The numerical amounts can be referred to as in dollars or pounds, or which ever currency most appropriate for you, but for now I will talk in terms of British pounds sterling.
In this initial scenario, the contestant has taken out 11 boxes, including the £20,000, the £35,000, and the £100,000 but it is still a reasonably good board. The banker now makes his highest offer so far of a respectable £12,500. The contestant declines the deal, a sensible choice based upon the odds, and elects to play on taking out the £75,000, an obvious disappointment, but then the 1p goes to everyone's relief except the banker's, and the final of these 3 boxes selected is the £5,000, by no means a disaster in gameplay terms at this stage, and these are the 8 boxes left:
Let's assume that the banker now offers £17,500 for the contestant to deal. Is it worth not dealing, and consequently going on? In my view no, based upon the odds of 64% that one of the two big numbers will be selected in the next 3 boxes. Should that happen, then the banker's offer is not likely to increase, and if anything it would be lower. For some the risk might be worth taking for the opportunity of going home with a life changing amount of money, but if you play the game sensibly with a combination of calculated risk and caution, you have a greater chance of beating the banker! Let's assume then, that the contestant plays on, no dealing amidst almost unbearable tension, and takes out the £100, the £750, and the £50,000 safety net, leaving the £250,000 very exposed but still in place. Here's what's left:
The banker is not sure about the contestant who has given the impression of being a bit of a gambler! His offer is lower, back at £12,500 to deal. On the face of it, with 5 boxes left, you would think that a fairer offer to make might be at least £25,000 yet the banker has probably calculated that in the next 3 boxes the contestant has an odds-on chance of taking out the £250,000, 60% to be precise, and there is no significant safety net. So the smart move is to deal at this stage, unless the contestant is already very wealthy and the loss of £12,500 would not be too great! The contestant chooses to play on in our hypothetical game with the lure of a possible quarter of a million pounds. To great cheers the first of 3 boxes they pick off is the £10, so they now have a 50% chance of avoiding the £250,000 in the next 2 boxes. Sadly the contestant picks it out in the very next box, and it is scant consolation that the final of their 3 boxes selected is the £1. So here's what's left:
Let's consider another scenario where there are 8 boxes left:
The banker offers £15,000 to deal. I think it is worth going on this time because there are three significant sums of money left, and only a 1.8% chance of taking them all out, so I would favour no deal. Let's assume the contestant does no deal, and takes out the £1, an excellent start, the £500 to a crescendo of audience delight, but then the good run ends when the final of the three boxes taken out is the £50,000. Here's what's left:
The banker now offers £20,000 to deal. I think it's worth dealing now because there is a 60% chance of taking out the £250,000 and a 30% chance of walking away with a relatively very small sum of money. The contestant deals, and then the post-deal openings vindicate their decision, because although they pick £10 and £100 first, the third box opened is the £250,000. Here is what's left:
The banker calls to say he would have offered £6,500 to deal which the contestant says they would not have accepted but gone on, and are relieved to find that £750 was in their box all along. Let's look at another scenario where when there are 5 boxes left, there are still two significant amounts of money:
The banker offers £30,000 for the contestant to leave at this point. There is a 30% chance that they will take out both the big numbers. They choose to play on in the hope of keeping the £250,000 in play. The very next box they pick to gasps from the audience is the £250,000. Now there is a 50% chance that the most they can walk away with at the very end is £500. It is all about the £50,000. The next box opened is £10. The tension is palpable. The final of the 3 boxes is revealed to contain just £1. So here is what's left:
Let's look at another scenario where when there are 5 boxes left, there are still three significant amounts of money:
The banker's offer at this stage is £25,000 to deal. Although there is an element of risk, the statistical odds are in favour of going on, so the contestant no deals. It is worth no dealing because there is only a 10% chance of disaster in opening the next 3 boxes! The £15,000 is taken out first which gets a mixed response from the audience. The 10p is chosen, and then amazingly the £50 is the last of the three to go. An almost dream scenario is left:
The banker now offers the contestant £120,000 to deal, and adds that he respects the contestant. The response is no deal. The contestant is prepared to gamble because they would be quite happy to walk away with £50,000. They have a 50% chance of losing £70,000, and a 50% chance of gaining £130,000 on the deal. It is because they will walk away with a significant amount of money whatever happens that makes it easier to go on to the end. The contestant is offered the opportunity by the banker of swapping the boxes but they decline, and open up their box to find: For me Deal Or No Deal is fascinating because it is all about nerve, the psychology between the contestant and the banker, and perfect timing!
There are 1,124,000,727,777,607,680,000 permutations of picking the 22 boxes. To achieve the perfect game of picking the 1p in ascending order sequentially up to £100,000, and thereby win £250,000 at the end would be a 1 in 1,124,000,727,777,607,680,000 chance. You would statistically expect that to happen once in 40 quadrillion centuries!
Here are some stats that you can apply on any given day about the probability of £250,000 being on the contestant's table. You can also swap the £250,000 for the 1p in this analysis if you want to know when the penny might next be sitting on the pound table!
The odds of the £250,000 appearing in the next show: 1/22 or 4.5%
The odds of the £250,000 appearing at least once in the next 6 shows = 24.4%
The odds of the £250,000 appearing at least once in the next 15 shows = 50.2%
The odds of the £250,000 appearing at least once in the next 30 shows = 75.32%
The odds of the £250,000 appearing at least once in the next 50 shows = 90.23.%
The odds of the £250,000 appearing at least once in the next 80 shows = 97.6.%
The odds of the £250,000 appearing at least once in the next 100 shows = 99%
The odds of the £250,000 appearing at least once in the next 110 shows = 99.4%
The odds of picking off the first 17 boxes, and see the final five boxes left containing none of the five highest value amounts of money left, the so called power five of £35,000 - £250,000, is 23.5%
The odds of going to the final 5 boxes with no number higher than £5,000 is 7.5%, and the odds of going to the final 8 boxes with no number higher than £5,000 is 1%
Money can change your life. Spiritual truth can change your eternal destiny. What are the odds based upon real evidence that the claims Jesus made about Himself are true? See
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