### FAQ: What are Las Vegas odds?

When statistical types use the word odds they mean the probability that an event will occur divided by the probability that an event will not occur, that is, odds = p/(1-p). However, in everyday conversation, people can use the word odds with a slightly different but related meaning. In ordinary conversation people often mean Las Vegas or gambling odds. So you will hear terms like 3to1 or 3to2. You may also hear people talk about these matters in terms of chance, as in, the chance is 3in4 or 3in5. So what do people mean when they say odds are 3to1 or the chance is 3in4?

Consider a probability of .5 (1/2 = .5) which is equivalent to the statistical odds of 1 (.5/(1-.5) = .5/.5 =1). In Las Vegas odds this would be given as 1to1, that is, one chance for to one chance against. When expressed as chance, it would be 1in2, that is, one chance in 2 of winning.

Now when a gambler says that the odds are 3to1, they mean that there are 3 chances to win to 1 chance to lose. Alternatively, the gambler express this as the chance is 3in4, that is, 3 chances to win in 4 total chances. What they really mean is that the probability is .75 (3/4 = .75) or that the statistical odds are 3 (odds = .75/(1-.75) = .75/.25 = 3).

You will also see Las Vegas odds of 3to2 (which is equivalent to 1.5to1, but you will never see it written this way). Odds of 3to2 is just another way of saying that there are 3 chances of winning to 2 chances of losing. The gambler may also express odds of 3to2 as a chance of 3in5, that is, 3 chances to win in 5 total chances. Whether given as odds of 3to2 or chance of 3in5 the statistical probability is .6 (3/5 = .6) and the statistical odds are 1.5 (odds = .6/(1-.6) = .6/.4 = 1.5).

The table below gives the Las Vegas (gambling) odds, chance, statistical odds and probability for a number of different combinations.

Table of Gambling Odds, Chance, Statistical Odds and Probability
vegas  chance     stat      prob
odds             odds
10to1  10in11  10.0000    0.9091

9to1   9in10   9.0000    0.9000
8to1   8in9    8.0000    0.8889
7to1   7in8    7.0000    0.8750
6to1   6in7    6.0000    0.8571
5to1   5in6    5.0000    0.8333

4to1   4in5    4.0000    0.8000
3to1   3in4    3.0000    0.7500
7to3   7in10   2.3333    0.7000
2to1   2in3    2.0000    0.6667
3to2   3in5    1.5000    0.6000

1to1   1in2    1.0000    0.5000

2to3   2in5    0.6667    0.4000
1to2   1in3    0.5000    0.3333
3to7   3in10   0.4286    0.3000
1to3   1in4    0.3333    0.2500
1to4   1in5    0.2500    0.2000

1to5   1in6    0.2000    0.1667
1to6   1in7    0.1667    0.1429
1to7   1in8    0.1429    0.1250
1to8   1in9    0.1250    0.1111
1to9   1in10   0.1111    0.1000

1to10  1in11   0.1000    0.0909

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