Correct spelling, explanation: likelihood is the correct spelling because it is a noun made by combining the word likely and suffix -hood. In the case of a word ending with y and a consonant before it, y transforms to i. Therefore likelihood is correct and likelyhood is incorrect. y would remain if there was a vowel before it.

Definition of likelihood:
noun, probability, a state of being likely
The likelihood of him being chosen as the team captain was very high.
If you work hard, the likelihood of success is increasing.

Incorrect spelling, explanation: this form is incorrect because y at the end of the word transforms to i every time any suffix is added, provided y is preceded by a consonant. Here, l is a consonant, so y becomes i and likelihood is the correct spelling. Likelyhood is incorrect because y would remain in the case of a vowel before y.