# Different between "X % more likely" vs. "X times more likely"

#### grreee12

##### New Member
I often see something like "smokers are 10% more likely to get cancer than non-smokers."

Is this equivalent to saying smokers are "10 times more likely"?

I think this is a probability vs. odds vs. percent increase thing, right?

Say we have a 2x2 table:

Code:
               Cancer  No Cancer  Total
Smokers        100      50        150
Non-Smokers    20       100       120
Probability = the fraction of times you expect the outcome in many trials
Odds = the probability the event occurring / the probability of event not occurring

I've tried to summarize some basic conclusions you could make from this data. I'm pretty sure the probability and odds stuff is correct, but I'm not sure about the last two headings where I compare the two groups using odds ratios and %s. Especially the very last bullet.

Any corrections needed? Thank you!

For Smokers:

1. The probability of smokers getting cancer = 100/150 or .67 or 67%
2. The probability of smokers NOT getting cancer = 50/150 or .33 or 33%
3. The odds of smokers getting cancer = .67 / .33 = 2.03.

Another way of saying #3 is the odds of getting cancer are 2.03 to 1.

(Can't #3 be calculated quicker by simply taking the number of smokers who get cancer divided by the number of smokers who didn't? (100/50 = 2.0).

For Non-Smokers:

1. The probability of non-smokers getting cancer = 20/120 or .17 or 17%
2. The probability of non-smokers NOT getting cancer = 100/120 or .83 or 83%
3. The odds of non-smokers getting cancer = .17 / .83 = 0.2.

Compare the two groups, using Odds Ratios:

Using Odds Ratio = odds of the 1st group / odds of the 2nd group

1. The ratio of the odds for smokers getting cancer to the odds for non-smokers is 2.03 / 0.2 or 10.15.
2. For smokers, the odds of getting cancer are:

• 10.15 times as large (i.e., 10.15 times larger than) as the odds for non-smokers getting cancer.
• 1,015% higher than the odds for non-smokers (if the OR was 1.15, it would be 15% higher; if 2.15, it would be 115% higher, etc.)

Compare the two groups, using raw percents:

• Smokers (67%) are more likely than non-smokers (17%) to get cancer.
• Smokers are 3.94 times (67%/17%) more likely than non-smokers to get cancer.
• Smokers are 34% (17% / (67%-17%) ) more likely than non-smokers to get cancer.

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