Most of the time “might” and “may” are almost interchangeable, with “might” suggesting a somewhat lower probability. You’re more likely to get wet if the forecaster says it may rain than if she says it might rain, but substituting one for the other is unlikely to get you into trouble—so long as you stay in the present tense.
But “might” is also the past tense of the auxiliary verb “may,” and is required in sentences like “Chuck might have avoided arrest for the robbery if he hadn’t given the teller his business card before asking for the money.” When speculating that events might have been other than they were, don’t substitute “may” for “might.”
When you are uncertain what has happened and are making a guess, then you may want to use “may”: “I think he may have thought I would really like an oil change for my birthday.”
As an aside: if you are an old-fashioned child, you will ask, “May I go out to play?” rather than “Can I go out to play?” Despite the prevalence of the latter pattern, some adults still feel strongly that “may” has to do with permission whereas “can” implies only physical ability. But then if you have a parent like this you’ve had this pattern drilled into your head long before you encountered this page.
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