p=55%,q=45%
(O means win, X means lose)
The probability of losing twice is qq.
Gambling has a situation of losing three times: OXX, and the probability is pqq.
In one case, four bets are lost: XOXX, probability pqqq.
There are three situations when you lose five times: XOXOX, XOOXX, OXOXX, and the probability is 3 ppqq.
Then the probability of losing is QQ+PQQ+PQQ+3PQQ.
No win or lose will only happen if you gamble four times: OOXX, the probability is ppqq.
Then the winning probability is1-(QQ+pqq+pqq+3pppqq+ppqq).
The result is 0.4920540625 (computer calculation).