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

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 117 times and lost 883 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 35 times and lost 65 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;
	}
}