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.
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