get the wrong end of the stick

get (hold of) the wrong end of the stick (to not understand a situation correctly) — понять превратно; не так понять; получить ошибочное представление

Example 1: She saw me together with her husband and got hold of the wrong end of the stick. — Она увидела меня вместе с её мужем и всё поняла не так.

Example 2: I said how nice he was and Julie got the wrong end of the stick and thought I wanted to go out with him. (CIDI)

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the short end (of the stick)

the short end (of the stick) (the smaller or less desirable part, rank, task, or amount) -- невыгодное, неблагоприятное положение; нечестная, несправедливая доля, отношение, положение, и т.п.; ~ несчастливый билет

get the short end of the stick (to suffer the bad effects of a situation; to feel that you are being treated particularly badly in comparison with other people )
не повезти / иметь невыгодное положение по сравнению с другими; ~ вытянуть несчастливый билет; ~ оказаться крайним; ~ оказаться тем, кому приходится расхлебывать кашу


.......The people who get the short end of the stick are those whose income is just too high to qualify for help from the government. (CALD)
When the mayor cut our budget almost in half, I felt like we were getting the short end of the stick.
..... I always feel like I'm getting the short end, that I'm being cheated out of something.
Cambridge Dictionary of American Idiom

. (*Typically: get ~; have ~; give someone ~; end up with ~.)
..... Why do I always get the short end of the stick? I want my fair share!
......She's unhappy because she has the short end of the stick again.
McGraw-Hill Dictionary of American Idioms and Phrasal Verbs.

He was given the short end of the stick — С ним обошлись несправедливо I was stuck with the short end of the stick again — Я снова оказался виноватым
I was left holding the short end of the stick — Расхлебывать кашу пришлось мне.

Origin
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Sometimes when there is an undesirable task to be completed by one member of a group, the individuals agree to "draw straws" (or in this case, sticks), which are broken in unequal lengths, then concealed in the hand of one person. The one who selects (draws) the shortest straw loses, and has to perform the task.
it means having bad luck, or getting a bad deal on something. it comes from a game called "drawing sticks", where a stick is broken into two uneven pieces, and each person picks a piece without looking, and whoever draws the shortest one loses (they usually have to do some unpleasant chore or task, such as walking the other persons dog, or paying for lunch, etc)
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It goes back in time when people drew sticks/ all but one being the same length and one short one. They were held in someones hand so no one could see the shortest one, who drew the short stick was stuck doing the job/task at hand.