Following footsteps: Terry Johnson and his daughter Alice on the family business

Alice Bailey Johnson is about to star as Marilyn Monroe in the revival production of Insignificance, written by her father, award-winning playwright and director Terry Johnson. Since Alice graduated from LAMDA she has worked on stage and screen, including several jobs working with with her dad as director. Here Johnson explains what it's like having your daughter decide to go into the family business, while Bailey Johnson reveals where her inspiration to become an actress came from.

Terry Johnson: Alice has always trod a very singular path and I think her desire to become an actress was announced to us. One of her friends applied to the BRIT school, which meant six of her friends applied and she was the one who got in. Her mum is an actress, and I'm a writer, but I think a ten minute chat was my only contribution to her training.

All our friends were in the theatre, so it was the world she grew up in. I was never worried about her deciding to be an actress. I didn't ever feel I needed to dissuade her from acting, which is what a lot of actors say about their kids. In my opinion you invest your life in your artform, it's nice to see them do it as well. It really didn't give me any anxiety.

I've never needed to offer advice to her. We've worked together a couple of times and it's always been very easy because she instinctively knows what work is to be done. She keeps herself well ahead of the game. She's a big thinker and she brews it – as soon as she's got a job she's brewing it. It's been a delight both times we've worked together.

Rehearsal rooms aren't a great place for dogs or children – a better place for dogs than children, in fact. So I don't think she spent much time in rehearsal rooms with me.

It was at her final LAMDA performance, where she played Sally Bowles from Cabaret when I knew she was definitely employable. She's a genuine triple threat. In terms of her career, the most proud I've been of her so far was in The Libertine. She was phenomenal in it. She's got incredible weight.

I wasn't involved in the casting process for this show, but it actually had occurred to me that the role of Marilyn is the right part for her. Traditionally, as the writer, you'd nip in and watch a run through, but Alice and I thought that it would be a waste of a run through. If anxiety rises then it's a pain in the bum.

Alice Bailey Johnson: I used to sit in dressing rooms backstage and act out the play I was listening to on the tannoy, so I think mum and dad's professions definitely influenced the path I have taken. I wouldn't have known about acting without them. Apparently I was in the car and I announced, probably very dramatically that: 'I don't want to do what you do daddy, I want to do what mummy does'.

I was exposed to so much theatre, TV and films but it was theatre I remember from a young age – sitting in lighting boxes and watching plays that went over my head. I just loved it, I wanted to be there all the time. But I also had a brilliant drama teacher called Mr Wooley. I went to an all girl's school and he was so enthusiastic and we did so many plays. We did Henry IV Part I and I just loved his ambition.

There was a point in an English lesson where we were given a list of some of the top British plays to read and dad's were on the list. I think I even asked if it was the Terry Johnson I thought it was. After that, I read all his stuff.

He lets me get on with it, but I can absolutely come to him about anything. He's been in the industry a long time and he knows how difficult it is and how tough it can be for actors. But he'll only ever offer advice up if I ask for it.

I think I was a bit apprehensive about working with him initially. You don't want people to presume that I'm only doing it because [he's the director or writer]. But as soon as I started, it was really normal. During The Libertine and Oh What a Lovely War he was a director first and my dad after. I think our relationship as actor and director has really blossomed. I felt really privileged to be able to do something together.

We hadn't talked about Insignificance previously and I had never read it, but when I did I realised it was a pretty stonking part. Dad hasn't been in on rehearsals. I think he would have if we had said: 'please do come in'. But I was quietly relieved that he wasn't in the rehearsal room. If the first time he saw it was on press night I think that might tip me over the edge, so he's seeing it on a preview. He trusts everyone.

Insignificance runs at the Arcola Theatre until 11 November.