Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 3 times and lost 7 times.

1. Car
2. Car
3. Goat
4. Car
5. Goat
6. Goat
7. Goat
8. Goat
9. Goat
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 329 times and lost 671 times.

1. Goat
2. Car
3. Car
4. Car
5. Goat
6. Goat
7. Car
8. Goat
9. Goat
10. Goat
11. Car
12. Goat
13. Goat
14. Goat
15. Goat
16. Car
17. Goat
18. Goat
19. Goat
20. Goat
21. Goat
22. Goat
23. Goat
24. Goat
25. Goat
26. Goat
27. Goat
28. Goat
29. Goat
30. Goat
31. Goat
32. Car
33. Car
34. Goat
35. Goat
36. Goat
37. Goat
38. Goat
39. Goat
40. Goat
41. Goat
42. Goat
43. Goat
44. Car
45. Goat
46. Goat
47. Goat
48. Goat
49. Goat
50. Goat
51. Goat
52. Goat
53. Goat
54. Goat
55. Goat
56. Goat
57. Goat
58. Goat
59. Goat
60. Car
61. Goat
62. Car
63. Goat
64. Goat
65. Goat
66. Goat
67. Goat
68. Car
69. Goat
70. Goat
71. Goat
72. Goat
73. Goat
74. Goat
75. Goat
76. Goat
77. Car
78. Car
79. Goat
80. Goat
81. Goat
82. Car
83. Goat
84. Car
85. Car
86. Goat
87. Car
88. Goat
89. Goat
90. Car
91. Goat
92. Goat
93. Car
94. Car
95. Car
96. Goat
97. Goat
98. Car
99. Goat
100. Car
101. Car
102. Car
103. Goat
104. Goat
105. Goat
106. Goat
107. Goat
108. Car
109. Goat
110. Goat
111. Goat
112. Goat
113. Goat
114. Goat
115. Goat
116. Car
117. Goat
118. Goat
119. Goat
120. Goat
121. Car
122. Car
123. Goat
124. Car
125. Goat
126. Car
127. Goat
128. Goat
129. Goat
130. Goat
131. Goat
132. Goat
133. Car
134. Goat
135. Goat
136. Car
137. Goat
138. Goat
139. Car
140. Car
141. Car
142. Goat
143. Goat
144. Goat
145. Goat
146. Goat
147. Goat
148. Goat
149. Goat
150. Goat
151. Car
152. Goat
153. Goat
154. Goat
155. Goat
156. Goat
157. Car
158. Goat
159. Goat
160. Goat
161. Goat
162. Car
163. Goat
164. Goat
165. Goat
166. Goat
167. Car
168. Car
169. Goat
170. Goat
171. Goat
172. Goat
173. Car
174. Car
175. Goat
176. Goat
177. Goat
178. Goat
179. Goat
180. Car
181. Goat
182. Goat
183. Goat
184. Car
185. Goat
186. Goat
187. Car
188. Goat
189. Goat
190. Goat
191. Goat
192. Car
193. Car
194. Goat
195. Goat
196. Goat
197. Car
198. Goat
199. Goat
200. Car
201. Car
202. Goat
203. Goat
204. Car
205. Car
206. Car
207. Goat
208. Car
209. Car
210. Goat
211. Goat
212. Goat
213. Car
214. Car
215. Goat
216. Goat
217. Goat
218. Car
219. Goat
220. Car
221. Goat
222. Goat
223. Goat
224. Goat
225. Goat
226. Car
227. Car
228. Goat
229. Car
230. Car
231. Goat
232. Goat
233. Car
234. Goat
235. Goat
236. Goat
237. Car
238. Goat
239. Car
240. Goat
241. Car
242. Goat
243. Car
244. Goat
245. Goat
246. Car
247. Goat
248. Goat
249. Goat
250. Car
251. Goat
252. Car
253. Goat
254. Goat
255. Goat
256. Goat
257. Goat
258. Car
259. Car
260. Goat
261. Goat
262. Car
263. Goat
264. Goat
265. Goat
266. Goat
267. Goat
268. Car
269. Goat
270. Goat
271. Goat
272. Goat
273. Goat
274. Goat
275. Goat
276. Goat
277. Goat
278. Goat
279. Car
280. Goat
281. Goat
282. Goat
283. Car
284. Goat
285. Car
286. Goat
287. Goat
288. Car
289. Goat
290. Goat
291. Goat
292. Goat
293. Goat
294. Goat
295. Car
296. Goat
297. Goat
298. Car
299. Goat
300. Car
301. Goat
302. Goat
303. Goat
304. Goat
305. Goat
306. Goat
307. Goat
308. Car
309. Goat
310. Goat
311. Goat
312. Goat
313. Goat
314. Goat
315. Goat
316. Goat
317. Goat
318. Goat
319. Car
320. Goat
321. Goat
322. Goat
323. Goat
324. Goat
325. Goat
326. Goat
327. Car
328. Goat
329. Goat
330. Goat
331. Car
332. Goat
333. Goat
334. Goat
335. Goat
336. Goat
337. Car
338. Goat
339. Car
340. Goat
341. Goat
342. Goat
343. Goat
344. Goat
345. Car
346. Goat
347. Goat
348. Car
349. Goat
350. Goat
351. Car
352. Car
353. Car
354. Goat
355. Car
356. Goat
357. Car
358. Car
359. Car
360. Goat
361. Car
362. Goat
363. Goat
364. Goat
365. Goat
366. Goat
367. Goat
368. Car
369. Car
370. Goat
371. Car
372. Goat
373. Goat
374. Goat
375. Goat
376. Goat
377. Goat
378. Car
379. Goat
380. Goat
381. Car
382. Car
383. Goat
384. Car
385. Goat
386. Goat
387. Goat
388. Goat
389. Goat
390. Goat
391. Goat
392. Goat
393. Car
394. Goat
395. Goat
396. Goat
397. Goat
398. Goat
399. Goat
400. Car
401. Car
402. Goat
403. Goat
404. Car
405. Goat
406. Car
407. Car
408. Goat
409. Car
410. Car
411. Car
412. Goat
413. Goat
414. Goat
415. Goat
416. Car
417. Car
418. Goat
419. Car
420. Goat
421. Goat
422. Goat
423. Goat
424. Car
425. Goat
426. Car
427. Goat
428. Car
429. Goat
430. Goat
431. Car
432. Car
433. Car
434. Goat
435. Car
436. Goat
437. Goat
438. Car
439. Car
440. Car
441. Goat
442. Car
443. Goat
444. Goat
445. Goat
446. Goat
447. Goat
448. Goat
449. Car
450. Goat
451. Car
452. Goat
453. Goat
454. Goat
455. Goat
456. Goat
457. Car
458. Goat
459. Goat
460. Goat
461. Goat
462. Car
463. Car
464. Car
465. Car
466. Goat
467. Car
468. Goat
469. Goat
470. Goat
471. Car
472. Goat
473. Goat
474. Goat
475. Car
476. Goat
477. Car
478. Goat
479. Car
480. Goat
481. Car
482. Goat
483. Goat
484. Car
485. Goat
486. Car
487. Car
488. Goat
489. Goat
490. Car
491. Goat
492. Goat
493. Goat
494. Car
495. Car
496. Goat
497. Car
498. Goat
499. Goat
500. Goat
501. Car
502. Goat
503. Goat
504. Goat
505. Goat
506. Car
507. Goat
508. Goat
509. Goat
510. Car
511. Goat
512. Car
513. Goat
514. Goat
515. Goat
516. Car
517. Car
518. Goat
519. Car
520. Car
521. Goat
522. Car
523. Goat
524. Goat
525. Car
526. Goat
527. Car
528. Goat
529. Goat
530. Goat
531. Goat
532. Car
533. Car
534. Goat
535. Car
536. Goat
537. Car
538. Goat
539. Car
540. Goat
541. Goat
542. Goat
543. Car
544. Car
545. Goat
546. Goat
547. Goat
548. Car
549. Car
550. Goat
551. Goat
552. Car
553. Goat
554. Goat
555. Goat
556. Goat
557. Goat
558. Car
559. Goat
560. Goat
561. Goat
562. Goat
563. Goat
564. Car
565. Goat
566. Goat
567. Car
568. Goat
569. Goat
570. Goat
571. Goat
572. Car
573. Goat
574. Car
575. Goat
576. Goat
577. Goat
578. Goat
579. Goat
580. Car
581. Car
582. Goat
583. Car
584. Goat
585. Goat
586. Car
587. Goat
588. Car
589. Car
590. Car
591. Goat
592. Car
593. Goat
594. Goat
595. Car
596. Goat
597. Car
598. Goat
599. Car
600. Car
601. Goat
602. Car
603. Car
604. Goat
605. Goat
606. Goat
607. Goat
608. Goat
609. Goat
610. Goat
611. Goat
612. Car
613. Goat
614. Car
615. Goat
616. Car
617. Car
618. Goat
619. Car
620. Goat
621. Car
622. Goat
623. Car
624. Car
625. Goat
626. Goat
627. Car
628. Car
629. Goat
630. Goat
631. Car
632. Goat
633. Goat
634. Goat
635. Goat
636. Goat
637. Goat
638. Goat
639. Goat
640. Goat
641. Goat
642. Car
643. Goat
644. Car
645. Car
646. Goat
647. Goat
648. Goat
649. Goat
650. Goat
651. Goat
652. Car
653. Car
654. Goat
655. Goat
656. Car
657. Goat
658. Goat
659. Goat
660. Goat
661. Car
662. Goat
663. Car
664. Car
665. Goat
666. Car
667. Goat
668. Goat
669. Car
670. Car
671. Goat
672. Goat
673. Goat
674. Goat
675. Goat
676. Goat
677. Car
678. Goat
679. Goat
680. Car
681. Goat
682. Car
683. Car
684. Goat
685. Goat
686. Car
687. Car
688. Car
689. Goat
690. Goat
691. Car
692. Goat
693. Car
694. Goat
695. Goat
696. Car
697. Car
698. Goat
699. Goat
700. Car
701. Goat
702. Goat
703. Goat
704. Car
705. Goat
706. Car
707. Goat
708. Goat
709. Goat
710. Goat
711. Car
712. Goat
713. Car
714. Goat
715. Goat
716. Car
717. Goat
718. Car
719. Goat
720. Car
721. Goat
722. Car
723. Goat
724. Goat
725. Car
726. Car
727. Goat
728. Car
729. Goat
730. Car
731. Car
732. Goat
733. Goat
734. Goat
735. Goat
736. Goat
737. Goat
738. Car
739. Goat
740. Car
741. Goat
742. Goat
743. Car
744. Car
745. Goat
746. Car
747. Goat
748. Goat
749. Car
750. Car
751. Goat
752. Goat
753. Car
754. Goat
755. Goat
756. Car
757. Car
758. Goat
759. Car
760. Goat
761. Goat
762. Goat
763. Goat
764. Car
765. Goat
766. Car
767. Car
768. Goat
769. Car
770. Goat
771. Car
772. Goat
773. Goat
774. Goat
775. Goat
776. Car
777. Goat
778. Goat
779. Goat
780. Goat
781. Car
782. Goat
783. Goat
784. Car
785. Car
786. Goat
787. Goat
788. Goat
789. Goat
790. Car
791. Goat
792. Goat
793. Goat
794. Car
795. Goat
796. Car
797. Goat
798. Goat
799. Goat
800. Goat
801. Goat
802. Car
803. Car
804. Car
805. Car
806. Car
807. Goat
808. Car
809. Goat
810. Goat
811. Car
812. Goat
813. Goat
814. Car
815. Goat
816. Goat
817. Goat
818. Car
819. Goat
820. Car
821. Car
822. Car
823. Goat
824. Goat
825. Goat
826. Goat
827. Car
828. Goat
829. Car
830. Goat
831. Car
832. Goat
833. Car
834. Goat
835. Car
836. Goat
837. Goat
838. Goat
839. Car
840. Goat
841. Goat
842. Goat
843. Goat
844. Car
845. Car
846. Goat
847. Goat
848. Goat
849. Goat
850. Goat
851. Goat
852. Goat
853. Goat
854. Goat
855. Car
856. Car
857. Goat
858. Goat
859. Goat
860. Car
861. Car
862. Goat
863. Car
864. Goat
865. Goat
866. Goat
867. Goat
868. Car
869. Goat
870. Car
871. Car
872. Goat
873. Goat
874. Car
875. Goat
876. Car
877. Goat
878. Car
879. Goat
880. Goat
881. Goat
882. Car
883. Goat
884. Car
885. Goat
886. Car
887. Goat
888. Goat
889. Goat
890. Goat
891. Car
892. Car
893. Goat
894. Goat
895. Goat
896. Goat
897. Goat
898. Goat
899. Goat
900. Goat
901. Goat
902. Car
903. Goat
904. Goat
905. Goat
906. Car
907. Goat
908. Car
909. Car
910. Goat
911. Goat
912. Goat
913. Car
914. Goat
915. Goat
916. Car
917. Goat
918. Goat
919. Goat
920. Goat
921. Goat
922. Goat
923. Goat
924. Car
925. Goat
926. Goat
927. Goat
928. Goat
929. Goat
930. Goat
931. Car
932. Car
933. Goat
934. Goat
935. Car
936. Goat
937. Goat
938. Goat
939. Car
940. Car
941. Goat
942. Goat
943. Goat
944. Car
945. Car
946. Goat
947. Car
948. Car
949. Goat
950. Car
951. Car
952. Car
953. Car
954. Car
955. Goat
956. Goat
957. Goat
958. Goat
959. Goat
960. Car
961. Car
962. Goat
963. Goat
964. Goat
965. Goat
966. Goat
967. Car
968. Car
969. Goat
970. Goat
971. Car
972. Goat
973. Goat
974. Goat
975. Goat
976. Goat
977. Goat
978. Goat
979. Goat
980. Goat
981. Goat
982. Goat
983. Goat
984. Car
985. Goat
986. Goat
987. Car
988. Car
989. Goat
990. Goat
991. Goat
992. Car
993. Car
994. Goat
995. Goat
996. Goat
997. Goat
998. Goat
999. Car
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 90 times and lost 910 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 39 times and lost 61 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}