Why do people think that Queen Mary of Scots was 1st cousin to elizabeth the first when in fact they have the same father king Henry the 8th which made the 2 queens half sisters?

No !! Mary Queen of Scots & Elizabeth do not have the same father ! Elizabeths 1/2 sister is Mary Tudor, Queen Mary I. Mary Queen of Scots is Mary Stuart, mother of James VI of Scotland, James I of England. She is Elizabeths cousin not her half sister. Mary I is Elizabeths 1/2 sister. Mary I & Mary, Queen of Scots are two (2) different people !