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Month of June has 30 days.

There are certainly 4 Saturdays (or any week day).
There are two more days.  They can be : Sat Sun, Sun Mon, Mon Tue, Tue Wed, Wed Thu, Thu Fri, Fri Sat.    So there are two out of seven possibilities in which Saturday occurs on 29th day or 30th day.

Hence, the probability = 2/7.

2 4 2
Number of days in june month =30
Consider 1st day of june is saturday .Then saturdays  occur on 8th , 15th ,22nd and finally on 28th.
Therefore there are five saturdays . 
The probability will be 5/30=1/6